Research ArticleThe Poacutelya-Szegouml Principle and the Anisotropic ConvexLorentz-Sobolev Inequality

Shuai Liu and Binwu He

Department of Mathematics Shanghai University Shanghai 200444 China

Correspondence should be addressed to Binwu He hebinwushueducn

Received 11 May 2014 Revised 24 June 2014 Accepted 24 June 2014 Published 15 July 2014

Academic Editor Seyed S Seyedalizadeh

Copyright copy 2014 S Liu and B He This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An anisotropic convex Lorentz-Sobolev inequality is established which extends Ludwig Xiao and Zhangrsquos result to any norm fromEuclidean norm and the geometric analogue of this inequality is given In addition it implies that the (anisotropic) Polya-Szegoprinciple is shown

1 Introduction

The classical Polya-Szego principle (see eg [1 2]) states thatfor 119901 ge 1 the inequality

intR119899

1003816100381610038161003816nabla1198911003816100381610038161003816119901119889119909 ge int

R119899

1003816100381610038161003816nabla119891⋆1003816100381610038161003816119901119889119909 (1)

holds for every 119891 isin 119862infin

0(R119899) where 119862infin

0(R119899) denotes the set

of functions onR119899 that are smooth and have compact supportand | sdot | is the standard Euclidean norm Here 119891⋆ denotes theSchwarz symmetrization of 119891 that is a function whose levelsets have the samemeasure as the level sets of119891 and are dilatesof the Euclidean unit ball 119861 It has important applicationsto a large class of variational problems in different areas forexample isoperimetric inequalities optimal forms of Sobolevinequalities and sharp a priori estimates of solutions tosecond-order elliptic or parabolic boundary value problems

An anisotropic version of the classical Polya-Szego prin-ciple has been proved in [3] where convex symmetrizationof 119891 is involved which states that if119870 is an origin-symmetriccompact convex set then for 119901 ge 1 the inequality

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 ge int

R119899

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909 (2)

holds for every 119891 isin 119862infin

0(R119899) where sdot 119870119900 is the Minkowski

functional of the polar body of 119870 Here 119891119870 denotes the

convex symmetrization of 119891 that is a function whose level

sets have the samemeasure as the level sets of119891 and are dilatesof the set 119870 Obviously (2) reduces to (1) when 119870 = 119861 (seeSection 2 for unexplained notation and terminology)

A new approach to understanding Polya-Szego principlewas proposed recently by Lutwak et al [4] and Zhang [5]Instead of using the classical technique on level sets [119891]

119905 their

approach is using the 119871119901 convexification of level sets ⟨119891⟩

119905

This technique plays a fundamental role in the newly emergedaffine Polya-Szego principle (see eg [4ndash9]) Despite thisprogress the study of the Polya-Szego principle by using thistechnique is vacancy This is the motivation of the presentpaper More precisely we show the Polya-Szego principlefrom the 119871

119901 Brunn-Minkowski theory different from theknown proofs of the Polya-Szego principle based on thegeometric measure theory (see eg [1ndash3 10ndash16])

In [17] Ludwig et al proved the following convexLorentz-Sobolev inequality (see Theorem 2 in [17]) if 119891 isin

119862infin

0(R119899) and 1 le 119901 lt 119899 then

intR119899

1003816100381610038161003816nabla1198911003816100381610038161003816119901119889119909 ge 119899120581

119901119899

119899int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905 (3)

where 119881 denotes the Lebesgue measure on R119899 with 120581119899 =

119881(119861) = 1205871198992

Γ(1 + 1198992) This inequality has a geometric ana-logue namely the following 119871119901 isoperimetric inequality for1 lt 119901 lt 119899

119878119901 (119871) ge 119899120581119901119899

119899119881(119871)(119899minus119901)119899

(4)

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 875245 6 pageshttpdxdoiorg1011552014875245

2 The Scientific World Journal

where119871 is an origin-symmetric compact convex set inR119899 and119878119901(119871) is the 119871

119901 surface area of 119871In this paper we establish the following anisotropic

convex Lorentz-Sobolev inequality

Theorem 1 If 119891 isin 119862infin

0(R119899) 1 le 119901 = 119899 and 119870 is an origin-

symmetric convex body in R119899 then

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 ge 119899119881(119870)

119901119899int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905 (5)

with equality if and only if ⟨119891⟩119905 is a dilate of119870 for almost every119905 gt 0

Similarly our inequality (5) has a geometric analoguenamely the following 119871

119901 Minkowski inequality for 1 lt 119901 lt

119899

119881119901 (119871 119870) ge 119881(119871)(119899minus119901)119899

119881(119870)119901119899

(6)

where 119871 119870 are origin-symmetric compact convex sets in R119899

and 119881119901(119871 119870) is the 119871119901 mixed volume of 119871 119870When 119871 = 119861 from 119878119901(119871) = 119899119881119901(119871 119861) (5) and (6) reduce

to (3) and (4) respectivelyIt is shown that our inequality (5) implies the anisotropic

Polya-Szego principle (2) for 1 le 119901 = 119899 in Theorem 5 Henceit is also true in Euclidean case that is (3) implies (1) for 1 le

119901 = 119899 The arguments after Theorem 5 yield the fact that theanisotropic Polya-Szego principle (2) is still true for 119901 = 119899

if we use the solution to the even normalized 119871119901 Minkowski

problem

2 Background Material

21 Elements of the119871119901 Brunn-MinkowskiTheory For later ref-erence we quickly recall in this subsection some backgroundmaterial from the 119871

119901 Brunn-Minkowski theory of convexbodies This theory has its origin in the work of Firey fromthe 1960s and has expanded rapidly over the last couple ofdecades (see eg [4 8 18ndash33])

A convex body is a compact convex set in R119899 whichis throughout assumed to contain the origin in its interiorWe denote by K119899

119900the space of convex bodies equipped

with the Hausdorff metric Each convex body 119870 is uniquelydetermined by its support function ℎ119870 = ℎ(119870 sdot) R119899 rarr R

defined by

ℎ119870 (119909) = ℎ (119870 119909) = max 119909 sdot 119910 119910 isin 119870 (7)

Let sdot 119870 R119899 rarr [0infin) denote the Minkowski functionalof 119870 isin K119899

119900 that is 119909119870 = min120582 ge 0 119909 isin 120582119870

The polar set 119870119900 of 119870 isin K119899119900is the convex body defined

by

119870119900= 119909 isin R

119899 119909 sdot 119910 le 1 forall119910 isin 119870 (8)

If 119870 isin K119899119900 then it follows from the definitions of support

functions and Minkowski functionals as well as the defini-tion of polar body that

ℎ119870 (sdot) = ℎ (119870 sdot) = sdot119870119900 (9)

For 119901 ge 1 119870 119871 isin K119899119900 the 119871119901 Minkowski combination

119870+119901119871 is the convex body defined by

ℎ(119870+119901119871 sdot)119901

= ℎ(119870 sdot)119901+ ℎ(119871 sdot)

119901 (10)

The 119871119901 mixed volume 119881119901(119870 119871) of 119870 119871 isin K119899119900is defined

in [25] by

119881119901 (119870 119871) =119901

119899lim120576rarr0+

119881(119870+1199011205761119901

119871) minus 119881 (119870)

120576 (11)

In particular

119881119901 (119870119870) = 119881 (119870) (12)

for every convex body119870It was shown in [25] that for all convex bodies119870 119871 isin K119899

119900

119881119901 (119870 119871) =1

119899int119878119899minus1

ℎ119901

119871(119906) 119889119878119901 (119870 119906) (13)

where 119878119901(119870 119906) = ℎ119870(119906)1minus119901

119889119878(119870 119906) and the measure 119878(119870 sdot)

on 119878119899minus1 is the classical surface area measure of119870 Recall that

for a Borel set 120596 sub 119878119899minus1 119878(119870 120596) is the (119899 minus 1)-dimensional

Hausdorff measure of the set of all boundary points of 119870 forwhich there exists a normal vector of119870 belonging to 120596

Note that

119878119901 (119905119870 sdot) = 119905119899minus119901

119878119901 (119870 sdot) (14)

for all 119905 gt 0 and convex bodies 119870

22 The Convex Symmetrization of Functions Given anymeasurable function 119891 R119899 rarr R such that 119881(119909 isin R119899

|119891(119909)| gt 119905) lt infin for every 119905 gt 0 its distribution function120583119891 [0infin) rarr [0infin] is defined by

120583119891 (119905) = 119881 (119909 isin R1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 gt 119905) (15)

The decreasing rearrangement 119891lowast [0infin) rarr [0infin] of 119891 isdefined by

119891lowast(119904) = inf 119905 ge 0 120583119891 (119905) le 119904 (16)

The Schwarz symmetrization of 119891 is the function 119891⋆ R119899 rarr

[0infin] defined by

119891⋆(119909) = 119891

lowast(120581119899|119909|

119899) (17)

where | sdot | is the standard Euclidean normFor an origin-symmetric convex body 119870 the convex

symmetrization 119891119870 of 119891 with respect to 119870 is defined as

follows

119891119870(119909) = 119891

lowast(120581119899119909

119899

) (18)

where 119909is the Minkowski functional of with being

a dilate of 119870 so that 119881() = 120581119899 Note that 119891 119891lowast and 119891

119870 areequimeasurable that is

120583119891 = 120583119891lowast = 120583119891119870 (19)

The Scientific World Journal 3

Therefore we have1003817100381710038171003817119891

1003817100381710038171003817infin = 119891lowast(0) =

1003817100381710038171003817100381711989111987010038171003817100381710038171003817infin

(20)

We will frequently apply Federerrsquos co-area formula (seeeg [34 page 258]) We state a version which is sufficient forour purposes if 119891 R119899 rarr R is Lipschitz and 119892 R119899 rarr

[0infin) is measurable then for any Borel set 119860 sube R

int119891minus1(119860)cap|nabla119891|gt0

119892 (119909) 119889119909 = int119860

int119891minus1119905

119892 (119909)1003816100381610038161003816nabla119891 (119909)

1003816100381610038161003816

119889H119899minus1

(119909) 119889119905

(21)

whereH119899minus1 denotes (119899minus1)-dimensional Hausdorffmeasure

23 The 119871119901 Convexification of Level Sets Suppose 119891 isin

119862infin

0(R119899) For each real 119905 gt 0 define the level set

[119891]119905= 119909 isin R

1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 ge 119905 (22)

By Sardrsquos theorem for almost every 119905 gt 0 the boundary

120597[119891]119905= 119909 isin R

1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 = 119905 (23)

of [119891]119905is a smooth (119899 minus 1)-dimensional submanifold of R119899

with everywhere nonzero normal vector nabla119891(119909)Now we explain the technique called the 119871119901 convexifica-

tion of level sets (see [17] for more details) Let 119891 119880 rarr Rwhere 119880 sub R119899 is open be locally Lipschitz let 119905 gt 0 andsuppose nabla119891(119909) = 0 for almost everywhere on 120597[119891]

119905= 119909 isin

119880 |119891(119909)| = 119905 For 1 le 119901 = 119899 define the 119871119901 convexification⟨119891⟩119905of the level set [119891]

119905as the unique origin-symmetric

convex body such that

int119878119899minus1

120593 (119906) 119889119878119901 (⟨119891⟩119905 119906) = int120597[119891]119905

120593 (] (119909)) 1003816100381610038161003816nabla1198911003816100381610038161003816119901minus1

119889H119899minus1

(119909)

(24)

for all even 120593 isin 119862(119878119899minus1

) where ](119909) = minusnabla119891(119909)|nabla119891(119909)|Thus equality (24) holds for almost every 119905 gt 0 if 119891 isin

119862infin

0(R119899)

3 The Anisotropic ConvexLorentz-Sobolev Inequality

The following lemma can be proved in the spirit of [17 3135](eg see Lemma 3 in [35])

Lemma 2 If 119891 isin 119862infin

0(R119899) and 119870 119871 are origin-symmetric

convex bodies then for almost every 119905 isin (0 119891infin) and 1 le

119901 = 119899 ⟨119891119870⟩119905is a dilate of 119870 and

119881(⟨119891119870⟩119905) = 119881(⟨119891

119871⟩119905) (25)

Proof Since ℎ119870119900 is Lipschitz (and therefore differentiablealmost everywhere) and ℎ119870119900(119909) = 1 on 120597119870 then for almostevery 119909 isin 120597119870

]119870 (119909) =nablaℎ119870119900 (119909)1003816100381610038161003816nablaℎ119870119900 (119909)

1003816100381610038161003816

(26)

where ]119870(119909) is the outer unit normal vector of119870 at the point119909 Note that ℎ119870(nablaℎ119870119900(119909)) = 1 for almost every 119909 isin R119899 hencewe have

ℎ119870 (]119870 (119909)) =1

1003816100381610038161003816nablaℎ119870119900 (119909)1003816100381610038161003816

(27)

Since119891119870 is Lipschitz then for almost every 119905 isin (0 119891infin) the

set 120597[119891119870]119905is the boundary of a dilate of119870 with nonvanishing

normal nabla119891119870 It follows from Sardrsquos theorem that

H119899(119909 isin R

1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 = 119905) = 0 for almost every 119905 gt 0

(28)

Hence there exists a unique 119904 gt 0 such that 119905 = 119891lowast(120581119899119904119899) for

almost every 119905 isin (0 119891infin) Indeed we have 119904 = (120583119891(119905)120581119899)

1119899Then by (24) (18) (9) and the fact thatnablaℎ

119900is homogeneous

of degree 0 and (27) we obtain that

int119878119899minus1

120593119901(119906) 119889119878119901 (⟨119891

119870⟩119905 119906)

= int120597[119891119870]

119905

120593119901(] (119909))

10038161003816100381610038161003816nabla119891119870(119909)

10038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909)

= int120597[119891119870]

119905

120593119901(] (119909))

times10038161003816100381610038161003816(119891lowast)1015840(120581119899ℎ119900(119909)

119899) 119899120581119899ℎ119900(119909)

119899minus1nablaℎ119900 (119909)

10038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909)

= int119904120597

120593119901(] (119909))

10038161003816100381610038161003816(119891lowast)1015840(120581119899119904119899) 119899120581119899119904

119899minus1nablaℎ119900 (119909)

10038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909)

= 119904119899minus1

(minus(119891lowast)1015840(120581119899119904119899) 119899120581119899119904

119899minus1)119901minus1

times int120597

120593119901(] (119909))

1003816100381610038161003816nablaℎ119900 (119909)1003816100381610038161003816119901minus1

119889H119899minus1

(119909)

= (120583119891 (119905)

120581119899)

(119899minus1)119899

(minus(119891lowast)1015840(120583119891 (119905)) 119899120581119899(

120583119891 (119905)

120581119899)

(119899minus1)119899

)

119901minus1

times int120597

120593119901(] (119909)) ℎ(] (119909))

1minus119901119889H119899minus1

(119909)

= 119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

times int119878119899minus1

120593119901(119906) 119889119878119901 ( 119906)

(29)

for almost every 119905 isin (0 119891infin) and even 120593 isin 119862(119878

119899minus1) Thus

the uniqueness of the solution of the even 119871119901 Minkowskiproblem [25] and (14) implies that

⟨119891119870⟩119905= 119888(119891 119905)

1(119899minus119901) for almost every 119905 isin (0

10038171003817100381710038171198911003817100381710038171003817infin)

(30)

4 The Scientific World Journal

where 119888(119891 119905) = 119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901 Since

119891119888119870

= 119891119870 for any 119888 gt 0 and any119870 isin K119899

119900 we have

119881(⟨119891119870⟩119905) = 119881(⟨119891

119871⟩119905) = 119888(119891 119905)

119899(119899minus119901)120581119899 (31)

for almost every 119905 isin (0 119891infin)

Recall that the 119871119901 Minkowski inequality [25] states the

following

Theorem 3 If 119901 ge 1 and 119871119870 isin K119899119900 then

119881119901 (119871 119870) ge 119881(119871)1minus119901119899

119881(119870)119901119899 (32)

with equality if and only if 119871 119870 are dilates when 119901 gt 1 and ifand only if 119871 119870 are homothetic when 119901 = 1

Now we prove the anisotropic convex Lorentz-Sobolevinequality

Proof of Theorem 1 Noting that ℎ119870(sdot) = sdot 119870119900 by the co-areaformula (21) (24) (13) and (32) we have

intR119899

ℎ119870(nabla119891)119901119889119909 = int

infin

0

int120597[119891]119905

ℎ119870(nabla119891)119901 11003816100381610038161003816nabla119891

1003816100381610038161003816

119889H119899minus1

(119909) 119889119905

= int

infin

0

int120597[119891]119905

ℎ119870(] (119909))1199011003816100381610038161003816nabla119891

1003816100381610038161003816119901minus1

119889H119899minus1

(119909) 119889119905

= int

infin

0

int119878119899minus1

ℎ119870(119906)119901119889119878119901 (⟨119891⟩119905 119906) 119889119905

= int

infin

0

119899119881119901 (⟨119891⟩119905 119870) 119889119905

ge 119899119881(119870)119901119899

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905

(33)

where ](119909) = minusnabla119891(119909)|nabla119891(119909)| on 120597[119891]119905for almost every 119905 gt 0

and the second equality holds since119870 is an origin-symmetricand the support function of 119870 is homogeneous of degree 1

Equality (5) follows from equality (32) and the fact that⟨119891⟩119905is origin-symmetric

It is shown above Proof of Theorem 1 that the 119871119901

Minkowski inequality (32) implies inequality (5)In what follows we will show that the 119871

119901 Minkowskiinequality (32) can be easily deduced from the anisotropicconvex Lorentz-Sobolev inequality (5) for 1 lt 119901 lt 119899 bytaking

119891 (119909) = 119892 (119909119871) where 119892 (119904) = (1 + 119904119901(119901minus1)

)1minus119899119901

(34)

Indeed as shown in [17 Lemma 8]

⟨119891⟩119905= 119888119901 (119905) 119871 (35)

and 119888119901(119905)119899minus119901

= |1198921015840(119904)|119901minus1

119904119899minus1 with 119905 = 119892(119904) Hence

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 = int

infin

0

119899119881119901 (⟨119891⟩119905 119870) 119889119905

= int

infin

0

119899119881119901 (119888119901 (119905) 119871 119870) 119889119905

= 119899119881 (119871119870)int

infin

0

119888119901(119905)119899minus119901

119889119905

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905 = int

infin

0

119881(119888119901 (119905) 119871)(119899minus119901)119899

119889119905

= 119881(119871)(119899minus119901)119899

int

infin

0

119888119901(119905)119899minus119901

119889119905

(36)

where

int

infin

0

119888119901(119905)119899minus119901

119889119905 =(119899 minus 119901)

119901

(119901 minus 1)119901minus1

119901119861(

119899 minus 119901

119901119899119901 minus 119899 + 119901

119901)

(37)

4 The Poacutelya-Szegouml Principle

The following theorem can be seen as a weak form of thePolya-Szego principle (2)

Theorem 4 If 119891 isin 119862infin

0(R119899) and 119870 119871 are origin-symmetric

convex body such that 119871 is not a dilate of119870 then for 1 le 119901 = 119899

intR119899

10038171003817100381710038171003817nabla11989111987110038171003817100381710038171003817

119901

119870119900119889119909 gt int

Rn

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909 (38)

Proof Since ⟨119891119871⟩119905is a dilate of 119871 for almost every 119905 isin

(0 119891infin) by Lemma 2 then the 119871

119901 Minkowski inequality(32) between ⟨119891

119871⟩119905and 119870 is strict for almost every 119905 isin

(0 119891infin) Combined with (25) it follows that

intR119899

10038171003817100381710038171003817nabla11989111987110038171003817100381710038171003817

119901

119870119900119889119909 = int

infin

0

119899119881119901 (⟨119891119871⟩119905 119870) 119889119905

gt 119899119881(119870)119901119899

int

infin

0

119881(⟨119891119871⟩119905)(119899minus119901)119899

119889119905

= 119899119881(119870)119901119899

int

infin

0

119881(⟨119891119870⟩119905)(119899minus119901)119899

119889119905

= int

infin

0

119899119881119901 (⟨119891119870⟩119905 119870) 119889119905

= intR119899

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909

(39)

We are now in the position to prove the Polya-Szegoprinciple (2)

Theorem 5 Suppose 119870 is an origin-symmetric convex bodiesin R119899 If 119891 isin 119862

infin

0(R119899) 1 le 119901 = 119899 then

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 ge int

R119899

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909 (40)

The Scientific World Journal 5

Proof It was shown in [4 (63)] that the following differentialinequality holds

119881(⟨119891⟩119905)(119899minus119901)119899

ge 119899119901minus1

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

(41)

Integrating both sides of the inequality gives

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905

ge 119899119901minus1

int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(42)

Noting that ℎ119870(sdot) = sdot 119870119900 and Combined with (5) we obtainthat

intR119899

ℎ119870(nabla119891)119901119889119909

ge 119899119901119881(119870)119901119899

int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(43)

By the homogeneous of 119870 in (43) and (40) we only need toconsider 119881(119870) = 120581119899 So it is sufficient to prove that

intR119899

ℎ119870(nabla119891119870)119901

119889119909 = 119899119901120581119901119899

119899int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(44)

The last equality is shown in [3]Nowwe prove this equal-ity by using Lemma 2 Together with the co-area formula (21)the equality (24) the definition of 119888(119891 119905) in Lemma 2 (13)and 119881(119870) = 120581119899 we obtain

intR119899

ℎ119870(nabla119891119870(119909))119901

119889119909

= int

infin

0

(int120597[119891119870]

119905

ℎ119870(nabla119891119870(119909))119901

1003816100381610038161003816nabla119891119870 (119909)

1003816100381610038161003816

119889H119899minus1

(119909))119889119905

= int

infin

0

int120597[119891119870]

119905

ℎ119870(] (119909))11990110038161003816100381610038161003816nabla11989111987010038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909) 119889119905

= int

infin

0

int119878119899minus1

ℎ119870(119906)119901119889119878119901 (⟨119891

119870⟩119905 119906) 119889119905

= int

infin

0

int119878119899minus1

119888 (119891 119905) ℎ119870(119906)119901119889119878119901 (119870 119906) 119889119905

= int

infin

0

119888 (119891 119905) 119889119905 int119878119899minus1

ℎ119870(119906)119901119889119878119901 (119870 119906)

= 119899119881 (119870)int

infin

0

119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

= 119899119901120581119901119899

119899int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(45)

where ](119909) = minusnabla119891119870(119909)|nabla119891

119870(119909)| on 120597[119891

119870]119905for almost every

119905 gt 0 And the second equality holds since 119870 is origin-symmetric and the support function of119870 is homogeneous ofdegree 1

MoreoverTheorem 5 can be proved for119901 ge 1 by using thesolution to the even normalized 119871119901Minkowski problem as in[7 9] More precisely suppose 119891 isin 119862

infin

0(R119899) for 119901 ge 1 and

define the normalized 119871119901 convexification ⟨119891⟩

119905as the unique

origin-symmetric convex body such that

1

119881 (⟨119891⟩119905)

int119878119899minus1

119892 (119906) 119889119878119901 (⟨119891⟩119905 119906)

= int120597[119891]119905

119892 (] (119909)) 1003816100381610038161003816nabla1198911003816100381610038161003816119901minus1

119889H119899minus1

(119909)

(46)

for almost every 119905 gt 0 By taking slight modifications in theproof of Theorem 1 we obtain

intR119899

ℎ119870(nabla119891)119901119889119909 ge 119899119881(119870)

119901119899int

infin

0

119881(⟨119891⟩119905)minus119901119899

119889119905 (47)

Similar to the proof ofTheorem 5 together with the observa-tion in [7 (422)] that

119881(⟨119891⟩119905)minus119901119899

ge 119899119901minus1

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

(48)

we also get (43) SoTheorem 5 remains true for 119901 = 119899

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All the authors contributed equally to the paper All theauthors read and approved the final paper

Acknowledgments

This work is supported in part by the National NaturalScience Foundation of China (Grant no 11371239) ShanghaiLeading Academic Discipline Project (Project no J50101)and the Research Fund for the Doctoral Programs of HigherEducation of China (20123108110001)

References

[1] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951

[2] G Talenti ldquoBest constant in Sobolev inequalityrdquo Annali diMatematica Pura ed Applicata Serie Quarta vol 110 pp 353ndash372 1976

[3] A Alvino V Ferone and G Trombetti ldquoConvex symmetriza-tion and applicationsrdquo Annales de lrsquoInstitut Henri Poincare CNon Linear Analysis vol 14 no 2 pp 275ndash293 1997

[4] E Lutwak D Yang and G Zhang ldquoSharp affine 119871119901 Sobolevinequalitiesrdquo Journal of Differential Geometry vol 62 no 1 pp17ndash38 2002

[5] G Zhang ldquoThe affine Sobolev inequalityrdquo Journal of DifferentialGeometry vol 53 no 1 pp 183ndash202 1999

6 The Scientific World Journal

[6] D Alonso-Gutierrez J Bastero and J Bernues ldquoFactoringSobolev inequalities through classes of functionsrdquo Proceedingsof the American Mathematical Society vol 140 no 10 pp 3557ndash3566 2012

[7] A Cianchi E Lutwak D Yang and G Zhang ldquoAffine Moser-Trudinger and MORrey-Sobolev inequalitiesrdquo Calculus of Vari-ations and Partial Differential Equations vol 36 no 3 pp 419ndash436 2009

[8] C Haberl and F E Schuster ldquoAsymmetric affine 119871119901 Sobolevinequalitiesrdquo Journal of Functional Analysis vol 257 no 3 pp641ndash658 2009

[9] C Haberl F Schuster and J Xiao ldquoAn asymmetric affine Polya-Szego principlerdquo Mathematische Annalen vol 352 no 3 pp517ndash542 2012

[10] J E Brothers and W P Ziemer ldquoMinimal rearrangementsof Sobolev functionsrdquo Journal fur die Reine und AngewandteMathematik vol 384 pp 153ndash179 1988

[11] A Cianchi L Esposito N Fusco and T Trombetti ldquoA quanti-tative Polya-Szego principlerdquo Journal fur die Reine und Ange-wandte Mathematik vol 614 pp 153ndash189 2008

[12] A Cianchi and N Fusco ldquoFunctions of bounded variation andrearrangementsrdquo Archive for Rational Mechanics and Analysisvol 165 no 1 pp 1ndash40 2002

[13] L Esposito and P Ronca ldquoQuantitative Polya-Szego principlefor convex symmetrizationrdquo Manuscripta Mathematica vol130 no 4 pp 433ndash459 2009

[14] L Esposito and C Trombetti ldquoConvex symmetrization andPolya-Szego inequalityrdquo Nonlinear Analysis Theory Methods ampApplications vol 56 no 1 pp 43ndash62 2004

[15] A Ferone and R Volpicelli ldquoMinimal rearrangements ofSobolev functions a new proofrdquo Annales de lrsquoInstitut HenriPoincare (C) Non Linear Analysis vol 20 no 2 pp 333ndash3392003

[16] A Ferone and R Volpicelli ldquoConvex rearrangement equalitycases in the Polya-Szego inequalityrdquo Calculus of Variations andPartial Differential Equations vol 21 no 3 pp 259ndash272 2004

[17] M Ludwig J Xiao and G Zhang ldquoSharp convex Lorentz-Sobolev inequalitiesrdquo Mathematische Annalen vol 350 no 1pp 169ndash197 2011

[18] S Campi and P Gronchi ldquoThe 119871119901-Busemann-Petty centroid

inequalityrdquoAdvances in Mathematics vol 167 no 1 pp 128ndash1412002

[19] S Campi and P Gronchi ldquoOn the reverse 119871119901-Busemann-Pettycentroid inequalityrdquo Mathematika vol 49 no 1-2 pp 1ndash11(2004) 2002

[20] C Haberl and F E Schuster ldquoGeneral L119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 83 no 1 pp1ndash26 2009

[21] D Hug E Lutwak D Yang and G Zhang ldquoOn the 119871119901 Minko-wski problem for polytopesrdquo Discrete amp Computational Geom-etry vol 33 no 4 pp 699ndash715 2005

[22] M Ludwig ldquoEllipsoids and matrix-valued valuationsrdquo DukeMathematical Journal vol 119 no 1 pp 159ndash188 2003

[23] M Ludwig ldquoMinkowski valuationsrdquo Transactions of the Ameri-can Mathematical Society vol 357 no 10 pp 4191ndash4213 2005

[24] M Ludwig and M Reitzner ldquoA classification of 119878119871(119899) invariantvaluationsrdquoAnnals ofMathematics vol 172 no 2 pp 1219ndash12672010

[25] E Lutwak ldquoThe Brunn-Minkowski-Firey theory I Mixedvolumes and the Minkowski problemrdquo Journal of DifferentialGeometry vol 38 no 1 pp 131ndash150 1993

[26] E Lutwak ldquoThe Brunn-Minkowski-Firey theory II Affine andgeominimal surface areasrdquo Advances in Mathematics vol 118no 2 pp 244ndash294 1996

[27] E Lutwak D Yang and G Zhang ldquo119871119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 56 no 1 pp111ndash132 2000

[28] E Lutwak D Yang and G Zhang ldquoA new ellipsoid associatedwith convex bodiesrdquoDukeMathematical Journal vol 104 no 3pp 375ndash390 2000

[29] E Lutwak D Yang and G Zhang ldquoOn the L119901-Minkowskiproblemrdquo Transactions of the American Mathematical Societyvol 356 no 11 pp 4359ndash4370 2004

[30] E Lutwak D Yang and G Zhang ldquo119871119901 John ellipsoidsrdquoProceedings of the London Mathematical Society vol 90 no 2pp 497ndash520 2005

[31] E Lutwak D Yang and G Zhang ldquoOptimal Sobolev normsand the 119871

119901 Minkowski problemrdquo International MathematicsResearch Notices vol 2006 21 pages 2006

[32] EWerner andD Ye ldquoNew 119871119901 affine isoperimetric inequalitiesrdquoAdvances in Mathematics vol 218 no 3 pp 762ndash780 2008

[33] E Werner and D Ye ldquoInequalities for mixed 119901-affine surfaceareardquoMathematische Annalen vol 347 no 3 pp 703ndash737 2010

[34] H Federer Geometric Measure Theory Springer Berlin Ger-many 1969

[35] T Wang ldquoThe affine Polya-Szego principle equality cases andstabilityrdquo Journal of Functional Analysis vol 265 no 8 pp 1728ndash1748 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

where119871 is an origin-symmetric compact convex set inR119899 and119878119901(119871) is the 119871

119901 surface area of 119871In this paper we establish the following anisotropic

convex Lorentz-Sobolev inequality

Theorem 1 If 119891 isin 119862infin

0(R119899) 1 le 119901 = 119899 and 119870 is an origin-

symmetric convex body in R119899 then

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 ge 119899119881(119870)

119901119899int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905 (5)

with equality if and only if ⟨119891⟩119905 is a dilate of119870 for almost every119905 gt 0

Similarly our inequality (5) has a geometric analoguenamely the following 119871

119901 Minkowski inequality for 1 lt 119901 lt

119899

119881119901 (119871 119870) ge 119881(119871)(119899minus119901)119899

119881(119870)119901119899

(6)

where 119871 119870 are origin-symmetric compact convex sets in R119899

and 119881119901(119871 119870) is the 119871119901 mixed volume of 119871 119870When 119871 = 119861 from 119878119901(119871) = 119899119881119901(119871 119861) (5) and (6) reduce

to (3) and (4) respectivelyIt is shown that our inequality (5) implies the anisotropic

Polya-Szego principle (2) for 1 le 119901 = 119899 in Theorem 5 Henceit is also true in Euclidean case that is (3) implies (1) for 1 le

119901 = 119899 The arguments after Theorem 5 yield the fact that theanisotropic Polya-Szego principle (2) is still true for 119901 = 119899

if we use the solution to the even normalized 119871119901 Minkowski

problem

2 Background Material

21 Elements of the119871119901 Brunn-MinkowskiTheory For later ref-erence we quickly recall in this subsection some backgroundmaterial from the 119871

119901 Brunn-Minkowski theory of convexbodies This theory has its origin in the work of Firey fromthe 1960s and has expanded rapidly over the last couple ofdecades (see eg [4 8 18ndash33])

A convex body is a compact convex set in R119899 whichis throughout assumed to contain the origin in its interiorWe denote by K119899

119900the space of convex bodies equipped

with the Hausdorff metric Each convex body 119870 is uniquelydetermined by its support function ℎ119870 = ℎ(119870 sdot) R119899 rarr R

defined by

ℎ119870 (119909) = ℎ (119870 119909) = max 119909 sdot 119910 119910 isin 119870 (7)

Let sdot 119870 R119899 rarr [0infin) denote the Minkowski functionalof 119870 isin K119899

119900 that is 119909119870 = min120582 ge 0 119909 isin 120582119870

The polar set 119870119900 of 119870 isin K119899119900is the convex body defined

by

119870119900= 119909 isin R

119899 119909 sdot 119910 le 1 forall119910 isin 119870 (8)

If 119870 isin K119899119900 then it follows from the definitions of support

functions and Minkowski functionals as well as the defini-tion of polar body that

ℎ119870 (sdot) = ℎ (119870 sdot) = sdot119870119900 (9)

For 119901 ge 1 119870 119871 isin K119899119900 the 119871119901 Minkowski combination

119870+119901119871 is the convex body defined by

ℎ(119870+119901119871 sdot)119901

= ℎ(119870 sdot)119901+ ℎ(119871 sdot)

119901 (10)

The 119871119901 mixed volume 119881119901(119870 119871) of 119870 119871 isin K119899119900is defined

in [25] by

119881119901 (119870 119871) =119901

119899lim120576rarr0+

119881(119870+1199011205761119901

119871) minus 119881 (119870)

120576 (11)

In particular

119881119901 (119870119870) = 119881 (119870) (12)

for every convex body119870It was shown in [25] that for all convex bodies119870 119871 isin K119899

119900

119881119901 (119870 119871) =1

119899int119878119899minus1

ℎ119901

119871(119906) 119889119878119901 (119870 119906) (13)

where 119878119901(119870 119906) = ℎ119870(119906)1minus119901

119889119878(119870 119906) and the measure 119878(119870 sdot)

on 119878119899minus1 is the classical surface area measure of119870 Recall that

for a Borel set 120596 sub 119878119899minus1 119878(119870 120596) is the (119899 minus 1)-dimensional

Hausdorff measure of the set of all boundary points of 119870 forwhich there exists a normal vector of119870 belonging to 120596

Note that

119878119901 (119905119870 sdot) = 119905119899minus119901

119878119901 (119870 sdot) (14)

for all 119905 gt 0 and convex bodies 119870

22 The Convex Symmetrization of Functions Given anymeasurable function 119891 R119899 rarr R such that 119881(119909 isin R119899

|119891(119909)| gt 119905) lt infin for every 119905 gt 0 its distribution function120583119891 [0infin) rarr [0infin] is defined by

120583119891 (119905) = 119881 (119909 isin R1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 gt 119905) (15)

The decreasing rearrangement 119891lowast [0infin) rarr [0infin] of 119891 isdefined by

119891lowast(119904) = inf 119905 ge 0 120583119891 (119905) le 119904 (16)

The Schwarz symmetrization of 119891 is the function 119891⋆ R119899 rarr

[0infin] defined by

119891⋆(119909) = 119891

lowast(120581119899|119909|

119899) (17)

where | sdot | is the standard Euclidean normFor an origin-symmetric convex body 119870 the convex

symmetrization 119891119870 of 119891 with respect to 119870 is defined as

follows

119891119870(119909) = 119891

lowast(120581119899119909

119899

) (18)

where 119909is the Minkowski functional of with being

a dilate of 119870 so that 119881() = 120581119899 Note that 119891 119891lowast and 119891

119870 areequimeasurable that is

120583119891 = 120583119891lowast = 120583119891119870 (19)

The Scientific World Journal 3

Therefore we have1003817100381710038171003817119891

1003817100381710038171003817infin = 119891lowast(0) =

1003817100381710038171003817100381711989111987010038171003817100381710038171003817infin

(20)

We will frequently apply Federerrsquos co-area formula (seeeg [34 page 258]) We state a version which is sufficient forour purposes if 119891 R119899 rarr R is Lipschitz and 119892 R119899 rarr

[0infin) is measurable then for any Borel set 119860 sube R

int119891minus1(119860)cap|nabla119891|gt0

119892 (119909) 119889119909 = int119860

int119891minus1119905

119892 (119909)1003816100381610038161003816nabla119891 (119909)

1003816100381610038161003816

119889H119899minus1

(119909) 119889119905

(21)

whereH119899minus1 denotes (119899minus1)-dimensional Hausdorffmeasure

23 The 119871119901 Convexification of Level Sets Suppose 119891 isin

119862infin

0(R119899) For each real 119905 gt 0 define the level set

[119891]119905= 119909 isin R

1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 ge 119905 (22)

By Sardrsquos theorem for almost every 119905 gt 0 the boundary

120597[119891]119905= 119909 isin R

1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 = 119905 (23)

of [119891]119905is a smooth (119899 minus 1)-dimensional submanifold of R119899

with everywhere nonzero normal vector nabla119891(119909)Now we explain the technique called the 119871119901 convexifica-

tion of level sets (see [17] for more details) Let 119891 119880 rarr Rwhere 119880 sub R119899 is open be locally Lipschitz let 119905 gt 0 andsuppose nabla119891(119909) = 0 for almost everywhere on 120597[119891]

119905= 119909 isin

119880 |119891(119909)| = 119905 For 1 le 119901 = 119899 define the 119871119901 convexification⟨119891⟩119905of the level set [119891]

119905as the unique origin-symmetric

convex body such that

int119878119899minus1

120593 (119906) 119889119878119901 (⟨119891⟩119905 119906) = int120597[119891]119905

120593 (] (119909)) 1003816100381610038161003816nabla1198911003816100381610038161003816119901minus1

119889H119899minus1

(119909)

(24)

for all even 120593 isin 119862(119878119899minus1

) where ](119909) = minusnabla119891(119909)|nabla119891(119909)|Thus equality (24) holds for almost every 119905 gt 0 if 119891 isin

119862infin

0(R119899)

3 The Anisotropic ConvexLorentz-Sobolev Inequality

The following lemma can be proved in the spirit of [17 3135](eg see Lemma 3 in [35])

Lemma 2 If 119891 isin 119862infin

0(R119899) and 119870 119871 are origin-symmetric

convex bodies then for almost every 119905 isin (0 119891infin) and 1 le

119901 = 119899 ⟨119891119870⟩119905is a dilate of 119870 and

119881(⟨119891119870⟩119905) = 119881(⟨119891

119871⟩119905) (25)

Proof Since ℎ119870119900 is Lipschitz (and therefore differentiablealmost everywhere) and ℎ119870119900(119909) = 1 on 120597119870 then for almostevery 119909 isin 120597119870

]119870 (119909) =nablaℎ119870119900 (119909)1003816100381610038161003816nablaℎ119870119900 (119909)

1003816100381610038161003816

(26)

where ]119870(119909) is the outer unit normal vector of119870 at the point119909 Note that ℎ119870(nablaℎ119870119900(119909)) = 1 for almost every 119909 isin R119899 hencewe have

ℎ119870 (]119870 (119909)) =1

1003816100381610038161003816nablaℎ119870119900 (119909)1003816100381610038161003816

(27)

Since119891119870 is Lipschitz then for almost every 119905 isin (0 119891infin) the

set 120597[119891119870]119905is the boundary of a dilate of119870 with nonvanishing

normal nabla119891119870 It follows from Sardrsquos theorem that

H119899(119909 isin R

1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 = 119905) = 0 for almost every 119905 gt 0

(28)

Hence there exists a unique 119904 gt 0 such that 119905 = 119891lowast(120581119899119904119899) for

almost every 119905 isin (0 119891infin) Indeed we have 119904 = (120583119891(119905)120581119899)

1119899Then by (24) (18) (9) and the fact thatnablaℎ

119900is homogeneous

of degree 0 and (27) we obtain that

int119878119899minus1

120593119901(119906) 119889119878119901 (⟨119891

119870⟩119905 119906)

= int120597[119891119870]

119905

120593119901(] (119909))

10038161003816100381610038161003816nabla119891119870(119909)

10038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909)

= int120597[119891119870]

119905

120593119901(] (119909))

times10038161003816100381610038161003816(119891lowast)1015840(120581119899ℎ119900(119909)

119899) 119899120581119899ℎ119900(119909)

119899minus1nablaℎ119900 (119909)

10038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909)

= int119904120597

120593119901(] (119909))

10038161003816100381610038161003816(119891lowast)1015840(120581119899119904119899) 119899120581119899119904

119899minus1nablaℎ119900 (119909)

10038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909)

= 119904119899minus1

(minus(119891lowast)1015840(120581119899119904119899) 119899120581119899119904

119899minus1)119901minus1

times int120597

120593119901(] (119909))

1003816100381610038161003816nablaℎ119900 (119909)1003816100381610038161003816119901minus1

119889H119899minus1

(119909)

= (120583119891 (119905)

120581119899)

(119899minus1)119899

(minus(119891lowast)1015840(120583119891 (119905)) 119899120581119899(

120583119891 (119905)

120581119899)

(119899minus1)119899

)

119901minus1

times int120597

120593119901(] (119909)) ℎ(] (119909))

1minus119901119889H119899minus1

(119909)

= 119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

times int119878119899minus1

120593119901(119906) 119889119878119901 ( 119906)

(29)

for almost every 119905 isin (0 119891infin) and even 120593 isin 119862(119878

119899minus1) Thus

the uniqueness of the solution of the even 119871119901 Minkowskiproblem [25] and (14) implies that

⟨119891119870⟩119905= 119888(119891 119905)

1(119899minus119901) for almost every 119905 isin (0

10038171003817100381710038171198911003817100381710038171003817infin)

(30)

4 The Scientific World Journal

where 119888(119891 119905) = 119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901 Since

119891119888119870

= 119891119870 for any 119888 gt 0 and any119870 isin K119899

119900 we have

119881(⟨119891119870⟩119905) = 119881(⟨119891

119871⟩119905) = 119888(119891 119905)

119899(119899minus119901)120581119899 (31)

for almost every 119905 isin (0 119891infin)

Recall that the 119871119901 Minkowski inequality [25] states the

following

Theorem 3 If 119901 ge 1 and 119871119870 isin K119899119900 then

119881119901 (119871 119870) ge 119881(119871)1minus119901119899

119881(119870)119901119899 (32)

with equality if and only if 119871 119870 are dilates when 119901 gt 1 and ifand only if 119871 119870 are homothetic when 119901 = 1

Now we prove the anisotropic convex Lorentz-Sobolevinequality

Proof of Theorem 1 Noting that ℎ119870(sdot) = sdot 119870119900 by the co-areaformula (21) (24) (13) and (32) we have

intR119899

ℎ119870(nabla119891)119901119889119909 = int

infin

0

int120597[119891]119905

ℎ119870(nabla119891)119901 11003816100381610038161003816nabla119891

1003816100381610038161003816

119889H119899minus1

(119909) 119889119905

= int

infin

0

int120597[119891]119905

ℎ119870(] (119909))1199011003816100381610038161003816nabla119891

1003816100381610038161003816119901minus1

119889H119899minus1

(119909) 119889119905

= int

infin

0

int119878119899minus1

ℎ119870(119906)119901119889119878119901 (⟨119891⟩119905 119906) 119889119905

= int

infin

0

119899119881119901 (⟨119891⟩119905 119870) 119889119905

ge 119899119881(119870)119901119899

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905

(33)

where ](119909) = minusnabla119891(119909)|nabla119891(119909)| on 120597[119891]119905for almost every 119905 gt 0

and the second equality holds since119870 is an origin-symmetricand the support function of 119870 is homogeneous of degree 1

Equality (5) follows from equality (32) and the fact that⟨119891⟩119905is origin-symmetric

It is shown above Proof of Theorem 1 that the 119871119901

Minkowski inequality (32) implies inequality (5)In what follows we will show that the 119871

119901 Minkowskiinequality (32) can be easily deduced from the anisotropicconvex Lorentz-Sobolev inequality (5) for 1 lt 119901 lt 119899 bytaking

119891 (119909) = 119892 (119909119871) where 119892 (119904) = (1 + 119904119901(119901minus1)

)1minus119899119901

(34)

Indeed as shown in [17 Lemma 8]

⟨119891⟩119905= 119888119901 (119905) 119871 (35)

and 119888119901(119905)119899minus119901

= |1198921015840(119904)|119901minus1

119904119899minus1 with 119905 = 119892(119904) Hence

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 = int

infin

0

119899119881119901 (⟨119891⟩119905 119870) 119889119905

= int

infin

0

119899119881119901 (119888119901 (119905) 119871 119870) 119889119905

= 119899119881 (119871119870)int

infin

0

119888119901(119905)119899minus119901

119889119905

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905 = int

infin

0

119881(119888119901 (119905) 119871)(119899minus119901)119899

119889119905

= 119881(119871)(119899minus119901)119899

int

infin

0

119888119901(119905)119899minus119901

119889119905

(36)

where

int

infin

0

119888119901(119905)119899minus119901

119889119905 =(119899 minus 119901)

119901

(119901 minus 1)119901minus1

119901119861(

119899 minus 119901

119901119899119901 minus 119899 + 119901

119901)

(37)

4 The Poacutelya-Szegouml Principle

The following theorem can be seen as a weak form of thePolya-Szego principle (2)

Theorem 4 If 119891 isin 119862infin

0(R119899) and 119870 119871 are origin-symmetric

convex body such that 119871 is not a dilate of119870 then for 1 le 119901 = 119899

intR119899

10038171003817100381710038171003817nabla11989111987110038171003817100381710038171003817

119901

119870119900119889119909 gt int

Rn

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909 (38)

Proof Since ⟨119891119871⟩119905is a dilate of 119871 for almost every 119905 isin

(0 119891infin) by Lemma 2 then the 119871

119901 Minkowski inequality(32) between ⟨119891

119871⟩119905and 119870 is strict for almost every 119905 isin

(0 119891infin) Combined with (25) it follows that

intR119899

10038171003817100381710038171003817nabla11989111987110038171003817100381710038171003817

119901

119870119900119889119909 = int

infin

0

119899119881119901 (⟨119891119871⟩119905 119870) 119889119905

gt 119899119881(119870)119901119899

int

infin

0

119881(⟨119891119871⟩119905)(119899minus119901)119899

119889119905

= 119899119881(119870)119901119899

int

infin

0

119881(⟨119891119870⟩119905)(119899minus119901)119899

119889119905

= int

infin

0

119899119881119901 (⟨119891119870⟩119905 119870) 119889119905

= intR119899

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909

(39)

We are now in the position to prove the Polya-Szegoprinciple (2)

Theorem 5 Suppose 119870 is an origin-symmetric convex bodiesin R119899 If 119891 isin 119862

infin

0(R119899) 1 le 119901 = 119899 then

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 ge int

R119899

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909 (40)

The Scientific World Journal 5

Proof It was shown in [4 (63)] that the following differentialinequality holds

119881(⟨119891⟩119905)(119899minus119901)119899

ge 119899119901minus1

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

(41)

Integrating both sides of the inequality gives

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905

ge 119899119901minus1

int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(42)

Noting that ℎ119870(sdot) = sdot 119870119900 and Combined with (5) we obtainthat

intR119899

ℎ119870(nabla119891)119901119889119909

ge 119899119901119881(119870)119901119899

int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(43)

By the homogeneous of 119870 in (43) and (40) we only need toconsider 119881(119870) = 120581119899 So it is sufficient to prove that

intR119899

ℎ119870(nabla119891119870)119901

119889119909 = 119899119901120581119901119899

119899int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(44)

The last equality is shown in [3]Nowwe prove this equal-ity by using Lemma 2 Together with the co-area formula (21)the equality (24) the definition of 119888(119891 119905) in Lemma 2 (13)and 119881(119870) = 120581119899 we obtain

intR119899

ℎ119870(nabla119891119870(119909))119901

119889119909

= int

infin

0

(int120597[119891119870]

119905

ℎ119870(nabla119891119870(119909))119901

1003816100381610038161003816nabla119891119870 (119909)

1003816100381610038161003816

119889H119899minus1

(119909))119889119905

= int

infin

0

int120597[119891119870]

119905

ℎ119870(] (119909))11990110038161003816100381610038161003816nabla11989111987010038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909) 119889119905

= int

infin

0

int119878119899minus1

ℎ119870(119906)119901119889119878119901 (⟨119891

119870⟩119905 119906) 119889119905

= int

infin

0

int119878119899minus1

119888 (119891 119905) ℎ119870(119906)119901119889119878119901 (119870 119906) 119889119905

= int

infin

0

119888 (119891 119905) 119889119905 int119878119899minus1

ℎ119870(119906)119901119889119878119901 (119870 119906)

= 119899119881 (119870)int

infin

0

119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

= 119899119901120581119901119899

119899int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(45)

where ](119909) = minusnabla119891119870(119909)|nabla119891

119870(119909)| on 120597[119891

119870]119905for almost every

119905 gt 0 And the second equality holds since 119870 is origin-symmetric and the support function of119870 is homogeneous ofdegree 1

MoreoverTheorem 5 can be proved for119901 ge 1 by using thesolution to the even normalized 119871119901Minkowski problem as in[7 9] More precisely suppose 119891 isin 119862

infin

0(R119899) for 119901 ge 1 and

define the normalized 119871119901 convexification ⟨119891⟩

119905as the unique

origin-symmetric convex body such that

1

119881 (⟨119891⟩119905)

int119878119899minus1

119892 (119906) 119889119878119901 (⟨119891⟩119905 119906)

= int120597[119891]119905

119892 (] (119909)) 1003816100381610038161003816nabla1198911003816100381610038161003816119901minus1

119889H119899minus1

(119909)

(46)

for almost every 119905 gt 0 By taking slight modifications in theproof of Theorem 1 we obtain

intR119899

ℎ119870(nabla119891)119901119889119909 ge 119899119881(119870)

119901119899int

infin

0

119881(⟨119891⟩119905)minus119901119899

119889119905 (47)

Similar to the proof ofTheorem 5 together with the observa-tion in [7 (422)] that

119881(⟨119891⟩119905)minus119901119899

ge 119899119901minus1

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

(48)

we also get (43) SoTheorem 5 remains true for 119901 = 119899

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All the authors contributed equally to the paper All theauthors read and approved the final paper

Acknowledgments

This work is supported in part by the National NaturalScience Foundation of China (Grant no 11371239) ShanghaiLeading Academic Discipline Project (Project no J50101)and the Research Fund for the Doctoral Programs of HigherEducation of China (20123108110001)

References

[1] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951

[2] G Talenti ldquoBest constant in Sobolev inequalityrdquo Annali diMatematica Pura ed Applicata Serie Quarta vol 110 pp 353ndash372 1976

[3] A Alvino V Ferone and G Trombetti ldquoConvex symmetriza-tion and applicationsrdquo Annales de lrsquoInstitut Henri Poincare CNon Linear Analysis vol 14 no 2 pp 275ndash293 1997

[4] E Lutwak D Yang and G Zhang ldquoSharp affine 119871119901 Sobolevinequalitiesrdquo Journal of Differential Geometry vol 62 no 1 pp17ndash38 2002

[5] G Zhang ldquoThe affine Sobolev inequalityrdquo Journal of DifferentialGeometry vol 53 no 1 pp 183ndash202 1999

6 The Scientific World Journal

[6] D Alonso-Gutierrez J Bastero and J Bernues ldquoFactoringSobolev inequalities through classes of functionsrdquo Proceedingsof the American Mathematical Society vol 140 no 10 pp 3557ndash3566 2012

[7] A Cianchi E Lutwak D Yang and G Zhang ldquoAffine Moser-Trudinger and MORrey-Sobolev inequalitiesrdquo Calculus of Vari-ations and Partial Differential Equations vol 36 no 3 pp 419ndash436 2009

[8] C Haberl and F E Schuster ldquoAsymmetric affine 119871119901 Sobolevinequalitiesrdquo Journal of Functional Analysis vol 257 no 3 pp641ndash658 2009

[9] C Haberl F Schuster and J Xiao ldquoAn asymmetric affine Polya-Szego principlerdquo Mathematische Annalen vol 352 no 3 pp517ndash542 2012

[10] J E Brothers and W P Ziemer ldquoMinimal rearrangementsof Sobolev functionsrdquo Journal fur die Reine und AngewandteMathematik vol 384 pp 153ndash179 1988

[11] A Cianchi L Esposito N Fusco and T Trombetti ldquoA quanti-tative Polya-Szego principlerdquo Journal fur die Reine und Ange-wandte Mathematik vol 614 pp 153ndash189 2008

[12] A Cianchi and N Fusco ldquoFunctions of bounded variation andrearrangementsrdquo Archive for Rational Mechanics and Analysisvol 165 no 1 pp 1ndash40 2002

[13] L Esposito and P Ronca ldquoQuantitative Polya-Szego principlefor convex symmetrizationrdquo Manuscripta Mathematica vol130 no 4 pp 433ndash459 2009

[14] L Esposito and C Trombetti ldquoConvex symmetrization andPolya-Szego inequalityrdquo Nonlinear Analysis Theory Methods ampApplications vol 56 no 1 pp 43ndash62 2004

[15] A Ferone and R Volpicelli ldquoMinimal rearrangements ofSobolev functions a new proofrdquo Annales de lrsquoInstitut HenriPoincare (C) Non Linear Analysis vol 20 no 2 pp 333ndash3392003

[16] A Ferone and R Volpicelli ldquoConvex rearrangement equalitycases in the Polya-Szego inequalityrdquo Calculus of Variations andPartial Differential Equations vol 21 no 3 pp 259ndash272 2004

[17] M Ludwig J Xiao and G Zhang ldquoSharp convex Lorentz-Sobolev inequalitiesrdquo Mathematische Annalen vol 350 no 1pp 169ndash197 2011

[18] S Campi and P Gronchi ldquoThe 119871119901-Busemann-Petty centroid

inequalityrdquoAdvances in Mathematics vol 167 no 1 pp 128ndash1412002

[19] S Campi and P Gronchi ldquoOn the reverse 119871119901-Busemann-Pettycentroid inequalityrdquo Mathematika vol 49 no 1-2 pp 1ndash11(2004) 2002

[20] C Haberl and F E Schuster ldquoGeneral L119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 83 no 1 pp1ndash26 2009

[21] D Hug E Lutwak D Yang and G Zhang ldquoOn the 119871119901 Minko-wski problem for polytopesrdquo Discrete amp Computational Geom-etry vol 33 no 4 pp 699ndash715 2005

[22] M Ludwig ldquoEllipsoids and matrix-valued valuationsrdquo DukeMathematical Journal vol 119 no 1 pp 159ndash188 2003

[23] M Ludwig ldquoMinkowski valuationsrdquo Transactions of the Ameri-can Mathematical Society vol 357 no 10 pp 4191ndash4213 2005

[24] M Ludwig and M Reitzner ldquoA classification of 119878119871(119899) invariantvaluationsrdquoAnnals ofMathematics vol 172 no 2 pp 1219ndash12672010

[25] E Lutwak ldquoThe Brunn-Minkowski-Firey theory I Mixedvolumes and the Minkowski problemrdquo Journal of DifferentialGeometry vol 38 no 1 pp 131ndash150 1993

[26] E Lutwak ldquoThe Brunn-Minkowski-Firey theory II Affine andgeominimal surface areasrdquo Advances in Mathematics vol 118no 2 pp 244ndash294 1996

[27] E Lutwak D Yang and G Zhang ldquo119871119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 56 no 1 pp111ndash132 2000

[28] E Lutwak D Yang and G Zhang ldquoA new ellipsoid associatedwith convex bodiesrdquoDukeMathematical Journal vol 104 no 3pp 375ndash390 2000

[29] E Lutwak D Yang and G Zhang ldquoOn the L119901-Minkowskiproblemrdquo Transactions of the American Mathematical Societyvol 356 no 11 pp 4359ndash4370 2004

[30] E Lutwak D Yang and G Zhang ldquo119871119901 John ellipsoidsrdquoProceedings of the London Mathematical Society vol 90 no 2pp 497ndash520 2005

[31] E Lutwak D Yang and G Zhang ldquoOptimal Sobolev normsand the 119871

119901 Minkowski problemrdquo International MathematicsResearch Notices vol 2006 21 pages 2006

[32] EWerner andD Ye ldquoNew 119871119901 affine isoperimetric inequalitiesrdquoAdvances in Mathematics vol 218 no 3 pp 762ndash780 2008

[33] E Werner and D Ye ldquoInequalities for mixed 119901-affine surfaceareardquoMathematische Annalen vol 347 no 3 pp 703ndash737 2010

[34] H Federer Geometric Measure Theory Springer Berlin Ger-many 1969

[35] T Wang ldquoThe affine Polya-Szego principle equality cases andstabilityrdquo Journal of Functional Analysis vol 265 no 8 pp 1728ndash1748 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

Therefore we have1003817100381710038171003817119891

1003817100381710038171003817infin = 119891lowast(0) =

1003817100381710038171003817100381711989111987010038171003817100381710038171003817infin

(20)

We will frequently apply Federerrsquos co-area formula (seeeg [34 page 258]) We state a version which is sufficient forour purposes if 119891 R119899 rarr R is Lipschitz and 119892 R119899 rarr

[0infin) is measurable then for any Borel set 119860 sube R

int119891minus1(119860)cap|nabla119891|gt0

119892 (119909) 119889119909 = int119860

int119891minus1119905

119892 (119909)1003816100381610038161003816nabla119891 (119909)

1003816100381610038161003816

119889H119899minus1

(119909) 119889119905

(21)

whereH119899minus1 denotes (119899minus1)-dimensional Hausdorffmeasure

23 The 119871119901 Convexification of Level Sets Suppose 119891 isin

119862infin

0(R119899) For each real 119905 gt 0 define the level set

[119891]119905= 119909 isin R

1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 ge 119905 (22)

By Sardrsquos theorem for almost every 119905 gt 0 the boundary

120597[119891]119905= 119909 isin R

1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 = 119905 (23)

of [119891]119905is a smooth (119899 minus 1)-dimensional submanifold of R119899

with everywhere nonzero normal vector nabla119891(119909)Now we explain the technique called the 119871119901 convexifica-

tion of level sets (see [17] for more details) Let 119891 119880 rarr Rwhere 119880 sub R119899 is open be locally Lipschitz let 119905 gt 0 andsuppose nabla119891(119909) = 0 for almost everywhere on 120597[119891]

119905= 119909 isin

119880 |119891(119909)| = 119905 For 1 le 119901 = 119899 define the 119871119901 convexification⟨119891⟩119905of the level set [119891]

119905as the unique origin-symmetric

convex body such that

int119878119899minus1

120593 (119906) 119889119878119901 (⟨119891⟩119905 119906) = int120597[119891]119905

120593 (] (119909)) 1003816100381610038161003816nabla1198911003816100381610038161003816119901minus1

119889H119899minus1

(119909)

(24)

for all even 120593 isin 119862(119878119899minus1

) where ](119909) = minusnabla119891(119909)|nabla119891(119909)|Thus equality (24) holds for almost every 119905 gt 0 if 119891 isin

119862infin

0(R119899)

3 The Anisotropic ConvexLorentz-Sobolev Inequality

The following lemma can be proved in the spirit of [17 3135](eg see Lemma 3 in [35])

Lemma 2 If 119891 isin 119862infin

0(R119899) and 119870 119871 are origin-symmetric

convex bodies then for almost every 119905 isin (0 119891infin) and 1 le

119901 = 119899 ⟨119891119870⟩119905is a dilate of 119870 and

119881(⟨119891119870⟩119905) = 119881(⟨119891

119871⟩119905) (25)

Proof Since ℎ119870119900 is Lipschitz (and therefore differentiablealmost everywhere) and ℎ119870119900(119909) = 1 on 120597119870 then for almostevery 119909 isin 120597119870

]119870 (119909) =nablaℎ119870119900 (119909)1003816100381610038161003816nablaℎ119870119900 (119909)

1003816100381610038161003816

(26)

where ]119870(119909) is the outer unit normal vector of119870 at the point119909 Note that ℎ119870(nablaℎ119870119900(119909)) = 1 for almost every 119909 isin R119899 hencewe have

ℎ119870 (]119870 (119909)) =1

1003816100381610038161003816nablaℎ119870119900 (119909)1003816100381610038161003816

(27)

Since119891119870 is Lipschitz then for almost every 119905 isin (0 119891infin) the

set 120597[119891119870]119905is the boundary of a dilate of119870 with nonvanishing

normal nabla119891119870 It follows from Sardrsquos theorem that

H119899(119909 isin R

1198991003816100381610038161003816119891 (119909)

1003816100381610038161003816 = 119905) = 0 for almost every 119905 gt 0

(28)

Hence there exists a unique 119904 gt 0 such that 119905 = 119891lowast(120581119899119904119899) for

almost every 119905 isin (0 119891infin) Indeed we have 119904 = (120583119891(119905)120581119899)

1119899Then by (24) (18) (9) and the fact thatnablaℎ

119900is homogeneous

of degree 0 and (27) we obtain that

int119878119899minus1

120593119901(119906) 119889119878119901 (⟨119891

119870⟩119905 119906)

= int120597[119891119870]

119905

120593119901(] (119909))

10038161003816100381610038161003816nabla119891119870(119909)

10038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909)

= int120597[119891119870]

119905

120593119901(] (119909))

times10038161003816100381610038161003816(119891lowast)1015840(120581119899ℎ119900(119909)

119899) 119899120581119899ℎ119900(119909)

119899minus1nablaℎ119900 (119909)

10038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909)

= int119904120597

120593119901(] (119909))

10038161003816100381610038161003816(119891lowast)1015840(120581119899119904119899) 119899120581119899119904

119899minus1nablaℎ119900 (119909)

10038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909)

= 119904119899minus1

(minus(119891lowast)1015840(120581119899119904119899) 119899120581119899119904

119899minus1)119901minus1

times int120597

120593119901(] (119909))

1003816100381610038161003816nablaℎ119900 (119909)1003816100381610038161003816119901minus1

119889H119899minus1

(119909)

= (120583119891 (119905)

120581119899)

(119899minus1)119899

(minus(119891lowast)1015840(120583119891 (119905)) 119899120581119899(

120583119891 (119905)

120581119899)

(119899minus1)119899

)

119901minus1

times int120597

120593119901(] (119909)) ℎ(] (119909))

1minus119901119889H119899minus1

(119909)

= 119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

times int119878119899minus1

120593119901(119906) 119889119878119901 ( 119906)

(29)

for almost every 119905 isin (0 119891infin) and even 120593 isin 119862(119878

119899minus1) Thus

the uniqueness of the solution of the even 119871119901 Minkowskiproblem [25] and (14) implies that

⟨119891119870⟩119905= 119888(119891 119905)

1(119899minus119901) for almost every 119905 isin (0

10038171003817100381710038171198911003817100381710038171003817infin)

(30)

4 The Scientific World Journal

where 119888(119891 119905) = 119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901 Since

119891119888119870

= 119891119870 for any 119888 gt 0 and any119870 isin K119899

119900 we have

119881(⟨119891119870⟩119905) = 119881(⟨119891

119871⟩119905) = 119888(119891 119905)

119899(119899minus119901)120581119899 (31)

for almost every 119905 isin (0 119891infin)

Recall that the 119871119901 Minkowski inequality [25] states the

following

Theorem 3 If 119901 ge 1 and 119871119870 isin K119899119900 then

119881119901 (119871 119870) ge 119881(119871)1minus119901119899

119881(119870)119901119899 (32)

with equality if and only if 119871 119870 are dilates when 119901 gt 1 and ifand only if 119871 119870 are homothetic when 119901 = 1

Now we prove the anisotropic convex Lorentz-Sobolevinequality

Proof of Theorem 1 Noting that ℎ119870(sdot) = sdot 119870119900 by the co-areaformula (21) (24) (13) and (32) we have

intR119899

ℎ119870(nabla119891)119901119889119909 = int

infin

0

int120597[119891]119905

ℎ119870(nabla119891)119901 11003816100381610038161003816nabla119891

1003816100381610038161003816

119889H119899minus1

(119909) 119889119905

= int

infin

0

int120597[119891]119905

ℎ119870(] (119909))1199011003816100381610038161003816nabla119891

1003816100381610038161003816119901minus1

119889H119899minus1

(119909) 119889119905

= int

infin

0

int119878119899minus1

ℎ119870(119906)119901119889119878119901 (⟨119891⟩119905 119906) 119889119905

= int

infin

0

119899119881119901 (⟨119891⟩119905 119870) 119889119905

ge 119899119881(119870)119901119899

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905

(33)

where ](119909) = minusnabla119891(119909)|nabla119891(119909)| on 120597[119891]119905for almost every 119905 gt 0

and the second equality holds since119870 is an origin-symmetricand the support function of 119870 is homogeneous of degree 1

Equality (5) follows from equality (32) and the fact that⟨119891⟩119905is origin-symmetric

It is shown above Proof of Theorem 1 that the 119871119901

Minkowski inequality (32) implies inequality (5)In what follows we will show that the 119871

119901 Minkowskiinequality (32) can be easily deduced from the anisotropicconvex Lorentz-Sobolev inequality (5) for 1 lt 119901 lt 119899 bytaking

119891 (119909) = 119892 (119909119871) where 119892 (119904) = (1 + 119904119901(119901minus1)

)1minus119899119901

(34)

Indeed as shown in [17 Lemma 8]

⟨119891⟩119905= 119888119901 (119905) 119871 (35)

and 119888119901(119905)119899minus119901

= |1198921015840(119904)|119901minus1

119904119899minus1 with 119905 = 119892(119904) Hence

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 = int

infin

0

119899119881119901 (⟨119891⟩119905 119870) 119889119905

= int

infin

0

119899119881119901 (119888119901 (119905) 119871 119870) 119889119905

= 119899119881 (119871119870)int

infin

0

119888119901(119905)119899minus119901

119889119905

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905 = int

infin

0

119881(119888119901 (119905) 119871)(119899minus119901)119899

119889119905

= 119881(119871)(119899minus119901)119899

int

infin

0

119888119901(119905)119899minus119901

119889119905

(36)

where

int

infin

0

119888119901(119905)119899minus119901

119889119905 =(119899 minus 119901)

119901

(119901 minus 1)119901minus1

119901119861(

119899 minus 119901

119901119899119901 minus 119899 + 119901

119901)

(37)

4 The Poacutelya-Szegouml Principle

The following theorem can be seen as a weak form of thePolya-Szego principle (2)

Theorem 4 If 119891 isin 119862infin

0(R119899) and 119870 119871 are origin-symmetric

convex body such that 119871 is not a dilate of119870 then for 1 le 119901 = 119899

intR119899

10038171003817100381710038171003817nabla11989111987110038171003817100381710038171003817

119901

119870119900119889119909 gt int

Rn

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909 (38)

Proof Since ⟨119891119871⟩119905is a dilate of 119871 for almost every 119905 isin

(0 119891infin) by Lemma 2 then the 119871

119901 Minkowski inequality(32) between ⟨119891

119871⟩119905and 119870 is strict for almost every 119905 isin

(0 119891infin) Combined with (25) it follows that

intR119899

10038171003817100381710038171003817nabla11989111987110038171003817100381710038171003817

119901

119870119900119889119909 = int

infin

0

119899119881119901 (⟨119891119871⟩119905 119870) 119889119905

gt 119899119881(119870)119901119899

int

infin

0

119881(⟨119891119871⟩119905)(119899minus119901)119899

119889119905

= 119899119881(119870)119901119899

int

infin

0

119881(⟨119891119870⟩119905)(119899minus119901)119899

119889119905

= int

infin

0

119899119881119901 (⟨119891119870⟩119905 119870) 119889119905

= intR119899

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909

(39)

We are now in the position to prove the Polya-Szegoprinciple (2)

Theorem 5 Suppose 119870 is an origin-symmetric convex bodiesin R119899 If 119891 isin 119862

infin

0(R119899) 1 le 119901 = 119899 then

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 ge int

R119899

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909 (40)

The Scientific World Journal 5

Proof It was shown in [4 (63)] that the following differentialinequality holds

119881(⟨119891⟩119905)(119899minus119901)119899

ge 119899119901minus1

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

(41)

Integrating both sides of the inequality gives

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905

ge 119899119901minus1

int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(42)

Noting that ℎ119870(sdot) = sdot 119870119900 and Combined with (5) we obtainthat

intR119899

ℎ119870(nabla119891)119901119889119909

ge 119899119901119881(119870)119901119899

int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(43)

By the homogeneous of 119870 in (43) and (40) we only need toconsider 119881(119870) = 120581119899 So it is sufficient to prove that

intR119899

ℎ119870(nabla119891119870)119901

119889119909 = 119899119901120581119901119899

119899int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(44)

The last equality is shown in [3]Nowwe prove this equal-ity by using Lemma 2 Together with the co-area formula (21)the equality (24) the definition of 119888(119891 119905) in Lemma 2 (13)and 119881(119870) = 120581119899 we obtain

intR119899

ℎ119870(nabla119891119870(119909))119901

119889119909

= int

infin

0

(int120597[119891119870]

119905

ℎ119870(nabla119891119870(119909))119901

1003816100381610038161003816nabla119891119870 (119909)

1003816100381610038161003816

119889H119899minus1

(119909))119889119905

= int

infin

0

int120597[119891119870]

119905

ℎ119870(] (119909))11990110038161003816100381610038161003816nabla11989111987010038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909) 119889119905

= int

infin

0

int119878119899minus1

ℎ119870(119906)119901119889119878119901 (⟨119891

119870⟩119905 119906) 119889119905

= int

infin

0

int119878119899minus1

119888 (119891 119905) ℎ119870(119906)119901119889119878119901 (119870 119906) 119889119905

= int

infin

0

119888 (119891 119905) 119889119905 int119878119899minus1

ℎ119870(119906)119901119889119878119901 (119870 119906)

= 119899119881 (119870)int

infin

0

119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

= 119899119901120581119901119899

119899int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(45)

where ](119909) = minusnabla119891119870(119909)|nabla119891

119870(119909)| on 120597[119891

119870]119905for almost every

119905 gt 0 And the second equality holds since 119870 is origin-symmetric and the support function of119870 is homogeneous ofdegree 1

MoreoverTheorem 5 can be proved for119901 ge 1 by using thesolution to the even normalized 119871119901Minkowski problem as in[7 9] More precisely suppose 119891 isin 119862

infin

0(R119899) for 119901 ge 1 and

define the normalized 119871119901 convexification ⟨119891⟩

119905as the unique

origin-symmetric convex body such that

1

119881 (⟨119891⟩119905)

int119878119899minus1

119892 (119906) 119889119878119901 (⟨119891⟩119905 119906)

= int120597[119891]119905

119892 (] (119909)) 1003816100381610038161003816nabla1198911003816100381610038161003816119901minus1

119889H119899minus1

(119909)

(46)

for almost every 119905 gt 0 By taking slight modifications in theproof of Theorem 1 we obtain

intR119899

ℎ119870(nabla119891)119901119889119909 ge 119899119881(119870)

119901119899int

infin

0

119881(⟨119891⟩119905)minus119901119899

119889119905 (47)

Similar to the proof ofTheorem 5 together with the observa-tion in [7 (422)] that

119881(⟨119891⟩119905)minus119901119899

ge 119899119901minus1

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

(48)

we also get (43) SoTheorem 5 remains true for 119901 = 119899

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All the authors contributed equally to the paper All theauthors read and approved the final paper

Acknowledgments

This work is supported in part by the National NaturalScience Foundation of China (Grant no 11371239) ShanghaiLeading Academic Discipline Project (Project no J50101)and the Research Fund for the Doctoral Programs of HigherEducation of China (20123108110001)

References

[1] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951

[2] G Talenti ldquoBest constant in Sobolev inequalityrdquo Annali diMatematica Pura ed Applicata Serie Quarta vol 110 pp 353ndash372 1976

[3] A Alvino V Ferone and G Trombetti ldquoConvex symmetriza-tion and applicationsrdquo Annales de lrsquoInstitut Henri Poincare CNon Linear Analysis vol 14 no 2 pp 275ndash293 1997

[4] E Lutwak D Yang and G Zhang ldquoSharp affine 119871119901 Sobolevinequalitiesrdquo Journal of Differential Geometry vol 62 no 1 pp17ndash38 2002

[5] G Zhang ldquoThe affine Sobolev inequalityrdquo Journal of DifferentialGeometry vol 53 no 1 pp 183ndash202 1999

6 The Scientific World Journal

[6] D Alonso-Gutierrez J Bastero and J Bernues ldquoFactoringSobolev inequalities through classes of functionsrdquo Proceedingsof the American Mathematical Society vol 140 no 10 pp 3557ndash3566 2012

[7] A Cianchi E Lutwak D Yang and G Zhang ldquoAffine Moser-Trudinger and MORrey-Sobolev inequalitiesrdquo Calculus of Vari-ations and Partial Differential Equations vol 36 no 3 pp 419ndash436 2009

[8] C Haberl and F E Schuster ldquoAsymmetric affine 119871119901 Sobolevinequalitiesrdquo Journal of Functional Analysis vol 257 no 3 pp641ndash658 2009

[9] C Haberl F Schuster and J Xiao ldquoAn asymmetric affine Polya-Szego principlerdquo Mathematische Annalen vol 352 no 3 pp517ndash542 2012

[10] J E Brothers and W P Ziemer ldquoMinimal rearrangementsof Sobolev functionsrdquo Journal fur die Reine und AngewandteMathematik vol 384 pp 153ndash179 1988

[11] A Cianchi L Esposito N Fusco and T Trombetti ldquoA quanti-tative Polya-Szego principlerdquo Journal fur die Reine und Ange-wandte Mathematik vol 614 pp 153ndash189 2008

[12] A Cianchi and N Fusco ldquoFunctions of bounded variation andrearrangementsrdquo Archive for Rational Mechanics and Analysisvol 165 no 1 pp 1ndash40 2002

[13] L Esposito and P Ronca ldquoQuantitative Polya-Szego principlefor convex symmetrizationrdquo Manuscripta Mathematica vol130 no 4 pp 433ndash459 2009

[14] L Esposito and C Trombetti ldquoConvex symmetrization andPolya-Szego inequalityrdquo Nonlinear Analysis Theory Methods ampApplications vol 56 no 1 pp 43ndash62 2004

[15] A Ferone and R Volpicelli ldquoMinimal rearrangements ofSobolev functions a new proofrdquo Annales de lrsquoInstitut HenriPoincare (C) Non Linear Analysis vol 20 no 2 pp 333ndash3392003

[16] A Ferone and R Volpicelli ldquoConvex rearrangement equalitycases in the Polya-Szego inequalityrdquo Calculus of Variations andPartial Differential Equations vol 21 no 3 pp 259ndash272 2004

[17] M Ludwig J Xiao and G Zhang ldquoSharp convex Lorentz-Sobolev inequalitiesrdquo Mathematische Annalen vol 350 no 1pp 169ndash197 2011

[18] S Campi and P Gronchi ldquoThe 119871119901-Busemann-Petty centroid

inequalityrdquoAdvances in Mathematics vol 167 no 1 pp 128ndash1412002

[19] S Campi and P Gronchi ldquoOn the reverse 119871119901-Busemann-Pettycentroid inequalityrdquo Mathematika vol 49 no 1-2 pp 1ndash11(2004) 2002

[20] C Haberl and F E Schuster ldquoGeneral L119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 83 no 1 pp1ndash26 2009

[21] D Hug E Lutwak D Yang and G Zhang ldquoOn the 119871119901 Minko-wski problem for polytopesrdquo Discrete amp Computational Geom-etry vol 33 no 4 pp 699ndash715 2005

[22] M Ludwig ldquoEllipsoids and matrix-valued valuationsrdquo DukeMathematical Journal vol 119 no 1 pp 159ndash188 2003

[23] M Ludwig ldquoMinkowski valuationsrdquo Transactions of the Ameri-can Mathematical Society vol 357 no 10 pp 4191ndash4213 2005

[24] M Ludwig and M Reitzner ldquoA classification of 119878119871(119899) invariantvaluationsrdquoAnnals ofMathematics vol 172 no 2 pp 1219ndash12672010

[25] E Lutwak ldquoThe Brunn-Minkowski-Firey theory I Mixedvolumes and the Minkowski problemrdquo Journal of DifferentialGeometry vol 38 no 1 pp 131ndash150 1993

[26] E Lutwak ldquoThe Brunn-Minkowski-Firey theory II Affine andgeominimal surface areasrdquo Advances in Mathematics vol 118no 2 pp 244ndash294 1996

[27] E Lutwak D Yang and G Zhang ldquo119871119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 56 no 1 pp111ndash132 2000

[28] E Lutwak D Yang and G Zhang ldquoA new ellipsoid associatedwith convex bodiesrdquoDukeMathematical Journal vol 104 no 3pp 375ndash390 2000

[29] E Lutwak D Yang and G Zhang ldquoOn the L119901-Minkowskiproblemrdquo Transactions of the American Mathematical Societyvol 356 no 11 pp 4359ndash4370 2004

[30] E Lutwak D Yang and G Zhang ldquo119871119901 John ellipsoidsrdquoProceedings of the London Mathematical Society vol 90 no 2pp 497ndash520 2005

[31] E Lutwak D Yang and G Zhang ldquoOptimal Sobolev normsand the 119871

119901 Minkowski problemrdquo International MathematicsResearch Notices vol 2006 21 pages 2006

[32] EWerner andD Ye ldquoNew 119871119901 affine isoperimetric inequalitiesrdquoAdvances in Mathematics vol 218 no 3 pp 762ndash780 2008

[33] E Werner and D Ye ldquoInequalities for mixed 119901-affine surfaceareardquoMathematische Annalen vol 347 no 3 pp 703ndash737 2010

[34] H Federer Geometric Measure Theory Springer Berlin Ger-many 1969

[35] T Wang ldquoThe affine Polya-Szego principle equality cases andstabilityrdquo Journal of Functional Analysis vol 265 no 8 pp 1728ndash1748 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

where 119888(119891 119905) = 119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901 Since

119891119888119870

= 119891119870 for any 119888 gt 0 and any119870 isin K119899

119900 we have

119881(⟨119891119870⟩119905) = 119881(⟨119891

119871⟩119905) = 119888(119891 119905)

119899(119899minus119901)120581119899 (31)

for almost every 119905 isin (0 119891infin)

Recall that the 119871119901 Minkowski inequality [25] states the

following

Theorem 3 If 119901 ge 1 and 119871119870 isin K119899119900 then

119881119901 (119871 119870) ge 119881(119871)1minus119901119899

119881(119870)119901119899 (32)

with equality if and only if 119871 119870 are dilates when 119901 gt 1 and ifand only if 119871 119870 are homothetic when 119901 = 1

Now we prove the anisotropic convex Lorentz-Sobolevinequality

Proof of Theorem 1 Noting that ℎ119870(sdot) = sdot 119870119900 by the co-areaformula (21) (24) (13) and (32) we have

intR119899

ℎ119870(nabla119891)119901119889119909 = int

infin

0

int120597[119891]119905

ℎ119870(nabla119891)119901 11003816100381610038161003816nabla119891

1003816100381610038161003816

119889H119899minus1

(119909) 119889119905

= int

infin

0

int120597[119891]119905

ℎ119870(] (119909))1199011003816100381610038161003816nabla119891

1003816100381610038161003816119901minus1

119889H119899minus1

(119909) 119889119905

= int

infin

0

int119878119899minus1

ℎ119870(119906)119901119889119878119901 (⟨119891⟩119905 119906) 119889119905

= int

infin

0

119899119881119901 (⟨119891⟩119905 119870) 119889119905

ge 119899119881(119870)119901119899

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905

(33)

where ](119909) = minusnabla119891(119909)|nabla119891(119909)| on 120597[119891]119905for almost every 119905 gt 0

and the second equality holds since119870 is an origin-symmetricand the support function of 119870 is homogeneous of degree 1

Equality (5) follows from equality (32) and the fact that⟨119891⟩119905is origin-symmetric

It is shown above Proof of Theorem 1 that the 119871119901

Minkowski inequality (32) implies inequality (5)In what follows we will show that the 119871

119901 Minkowskiinequality (32) can be easily deduced from the anisotropicconvex Lorentz-Sobolev inequality (5) for 1 lt 119901 lt 119899 bytaking

119891 (119909) = 119892 (119909119871) where 119892 (119904) = (1 + 119904119901(119901minus1)

)1minus119899119901

(34)

Indeed as shown in [17 Lemma 8]

⟨119891⟩119905= 119888119901 (119905) 119871 (35)

and 119888119901(119905)119899minus119901

= |1198921015840(119904)|119901minus1

119904119899minus1 with 119905 = 119892(119904) Hence

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 = int

infin

0

119899119881119901 (⟨119891⟩119905 119870) 119889119905

= int

infin

0

119899119881119901 (119888119901 (119905) 119871 119870) 119889119905

= 119899119881 (119871119870)int

infin

0

119888119901(119905)119899minus119901

119889119905

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905 = int

infin

0

119881(119888119901 (119905) 119871)(119899minus119901)119899

119889119905

= 119881(119871)(119899minus119901)119899

int

infin

0

119888119901(119905)119899minus119901

119889119905

(36)

where

int

infin

0

119888119901(119905)119899minus119901

119889119905 =(119899 minus 119901)

119901

(119901 minus 1)119901minus1

119901119861(

119899 minus 119901

119901119899119901 minus 119899 + 119901

119901)

(37)

4 The Poacutelya-Szegouml Principle

The following theorem can be seen as a weak form of thePolya-Szego principle (2)

Theorem 4 If 119891 isin 119862infin

0(R119899) and 119870 119871 are origin-symmetric

convex body such that 119871 is not a dilate of119870 then for 1 le 119901 = 119899

intR119899

10038171003817100381710038171003817nabla11989111987110038171003817100381710038171003817

119901

119870119900119889119909 gt int

Rn

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909 (38)

Proof Since ⟨119891119871⟩119905is a dilate of 119871 for almost every 119905 isin

(0 119891infin) by Lemma 2 then the 119871

119901 Minkowski inequality(32) between ⟨119891

119871⟩119905and 119870 is strict for almost every 119905 isin

(0 119891infin) Combined with (25) it follows that

intR119899

10038171003817100381710038171003817nabla11989111987110038171003817100381710038171003817

119901

119870119900119889119909 = int

infin

0

119899119881119901 (⟨119891119871⟩119905 119870) 119889119905

gt 119899119881(119870)119901119899

int

infin

0

119881(⟨119891119871⟩119905)(119899minus119901)119899

119889119905

= 119899119881(119870)119901119899

int

infin

0

119881(⟨119891119870⟩119905)(119899minus119901)119899

119889119905

= int

infin

0

119899119881119901 (⟨119891119870⟩119905 119870) 119889119905

= intR119899

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909

(39)

We are now in the position to prove the Polya-Szegoprinciple (2)

Theorem 5 Suppose 119870 is an origin-symmetric convex bodiesin R119899 If 119891 isin 119862

infin

0(R119899) 1 le 119901 = 119899 then

intR119899

1003817100381710038171003817nabla1198911003817100381710038171003817119901

119870119900119889119909 ge int

R119899

10038171003817100381710038171003817nabla11989111987010038171003817100381710038171003817

119901

119870119900119889119909 (40)

The Scientific World Journal 5

Proof It was shown in [4 (63)] that the following differentialinequality holds

119881(⟨119891⟩119905)(119899minus119901)119899

ge 119899119901minus1

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

(41)

Integrating both sides of the inequality gives

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905

ge 119899119901minus1

int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(42)

Noting that ℎ119870(sdot) = sdot 119870119900 and Combined with (5) we obtainthat

intR119899

ℎ119870(nabla119891)119901119889119909

ge 119899119901119881(119870)119901119899

int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(43)

By the homogeneous of 119870 in (43) and (40) we only need toconsider 119881(119870) = 120581119899 So it is sufficient to prove that

intR119899

ℎ119870(nabla119891119870)119901

119889119909 = 119899119901120581119901119899

119899int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(44)

The last equality is shown in [3]Nowwe prove this equal-ity by using Lemma 2 Together with the co-area formula (21)the equality (24) the definition of 119888(119891 119905) in Lemma 2 (13)and 119881(119870) = 120581119899 we obtain

intR119899

ℎ119870(nabla119891119870(119909))119901

119889119909

= int

infin

0

(int120597[119891119870]

119905

ℎ119870(nabla119891119870(119909))119901

1003816100381610038161003816nabla119891119870 (119909)

1003816100381610038161003816

119889H119899minus1

(119909))119889119905

= int

infin

0

int120597[119891119870]

119905

ℎ119870(] (119909))11990110038161003816100381610038161003816nabla11989111987010038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909) 119889119905

= int

infin

0

int119878119899minus1

ℎ119870(119906)119901119889119878119901 (⟨119891

119870⟩119905 119906) 119889119905

= int

infin

0

int119878119899minus1

119888 (119891 119905) ℎ119870(119906)119901119889119878119901 (119870 119906) 119889119905

= int

infin

0

119888 (119891 119905) 119889119905 int119878119899minus1

ℎ119870(119906)119901119889119878119901 (119870 119906)

= 119899119881 (119870)int

infin

0

119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

= 119899119901120581119901119899

119899int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(45)

where ](119909) = minusnabla119891119870(119909)|nabla119891

119870(119909)| on 120597[119891

119870]119905for almost every

119905 gt 0 And the second equality holds since 119870 is origin-symmetric and the support function of119870 is homogeneous ofdegree 1

MoreoverTheorem 5 can be proved for119901 ge 1 by using thesolution to the even normalized 119871119901Minkowski problem as in[7 9] More precisely suppose 119891 isin 119862

infin

0(R119899) for 119901 ge 1 and

define the normalized 119871119901 convexification ⟨119891⟩

119905as the unique

origin-symmetric convex body such that

1

119881 (⟨119891⟩119905)

int119878119899minus1

119892 (119906) 119889119878119901 (⟨119891⟩119905 119906)

= int120597[119891]119905

119892 (] (119909)) 1003816100381610038161003816nabla1198911003816100381610038161003816119901minus1

119889H119899minus1

(119909)

(46)

for almost every 119905 gt 0 By taking slight modifications in theproof of Theorem 1 we obtain

intR119899

ℎ119870(nabla119891)119901119889119909 ge 119899119881(119870)

119901119899int

infin

0

119881(⟨119891⟩119905)minus119901119899

119889119905 (47)

Similar to the proof ofTheorem 5 together with the observa-tion in [7 (422)] that

119881(⟨119891⟩119905)minus119901119899

ge 119899119901minus1

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

(48)

we also get (43) SoTheorem 5 remains true for 119901 = 119899

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All the authors contributed equally to the paper All theauthors read and approved the final paper

Acknowledgments

This work is supported in part by the National NaturalScience Foundation of China (Grant no 11371239) ShanghaiLeading Academic Discipline Project (Project no J50101)and the Research Fund for the Doctoral Programs of HigherEducation of China (20123108110001)

References

[1] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951

[2] G Talenti ldquoBest constant in Sobolev inequalityrdquo Annali diMatematica Pura ed Applicata Serie Quarta vol 110 pp 353ndash372 1976

[3] A Alvino V Ferone and G Trombetti ldquoConvex symmetriza-tion and applicationsrdquo Annales de lrsquoInstitut Henri Poincare CNon Linear Analysis vol 14 no 2 pp 275ndash293 1997

[4] E Lutwak D Yang and G Zhang ldquoSharp affine 119871119901 Sobolevinequalitiesrdquo Journal of Differential Geometry vol 62 no 1 pp17ndash38 2002

[5] G Zhang ldquoThe affine Sobolev inequalityrdquo Journal of DifferentialGeometry vol 53 no 1 pp 183ndash202 1999

6 The Scientific World Journal

[6] D Alonso-Gutierrez J Bastero and J Bernues ldquoFactoringSobolev inequalities through classes of functionsrdquo Proceedingsof the American Mathematical Society vol 140 no 10 pp 3557ndash3566 2012

[7] A Cianchi E Lutwak D Yang and G Zhang ldquoAffine Moser-Trudinger and MORrey-Sobolev inequalitiesrdquo Calculus of Vari-ations and Partial Differential Equations vol 36 no 3 pp 419ndash436 2009

[8] C Haberl and F E Schuster ldquoAsymmetric affine 119871119901 Sobolevinequalitiesrdquo Journal of Functional Analysis vol 257 no 3 pp641ndash658 2009

[9] C Haberl F Schuster and J Xiao ldquoAn asymmetric affine Polya-Szego principlerdquo Mathematische Annalen vol 352 no 3 pp517ndash542 2012

[10] J E Brothers and W P Ziemer ldquoMinimal rearrangementsof Sobolev functionsrdquo Journal fur die Reine und AngewandteMathematik vol 384 pp 153ndash179 1988

[11] A Cianchi L Esposito N Fusco and T Trombetti ldquoA quanti-tative Polya-Szego principlerdquo Journal fur die Reine und Ange-wandte Mathematik vol 614 pp 153ndash189 2008

[12] A Cianchi and N Fusco ldquoFunctions of bounded variation andrearrangementsrdquo Archive for Rational Mechanics and Analysisvol 165 no 1 pp 1ndash40 2002

[13] L Esposito and P Ronca ldquoQuantitative Polya-Szego principlefor convex symmetrizationrdquo Manuscripta Mathematica vol130 no 4 pp 433ndash459 2009

[14] L Esposito and C Trombetti ldquoConvex symmetrization andPolya-Szego inequalityrdquo Nonlinear Analysis Theory Methods ampApplications vol 56 no 1 pp 43ndash62 2004

[15] A Ferone and R Volpicelli ldquoMinimal rearrangements ofSobolev functions a new proofrdquo Annales de lrsquoInstitut HenriPoincare (C) Non Linear Analysis vol 20 no 2 pp 333ndash3392003

[16] A Ferone and R Volpicelli ldquoConvex rearrangement equalitycases in the Polya-Szego inequalityrdquo Calculus of Variations andPartial Differential Equations vol 21 no 3 pp 259ndash272 2004

[17] M Ludwig J Xiao and G Zhang ldquoSharp convex Lorentz-Sobolev inequalitiesrdquo Mathematische Annalen vol 350 no 1pp 169ndash197 2011

[18] S Campi and P Gronchi ldquoThe 119871119901-Busemann-Petty centroid

inequalityrdquoAdvances in Mathematics vol 167 no 1 pp 128ndash1412002

[19] S Campi and P Gronchi ldquoOn the reverse 119871119901-Busemann-Pettycentroid inequalityrdquo Mathematika vol 49 no 1-2 pp 1ndash11(2004) 2002

[20] C Haberl and F E Schuster ldquoGeneral L119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 83 no 1 pp1ndash26 2009

[21] D Hug E Lutwak D Yang and G Zhang ldquoOn the 119871119901 Minko-wski problem for polytopesrdquo Discrete amp Computational Geom-etry vol 33 no 4 pp 699ndash715 2005

[22] M Ludwig ldquoEllipsoids and matrix-valued valuationsrdquo DukeMathematical Journal vol 119 no 1 pp 159ndash188 2003

[23] M Ludwig ldquoMinkowski valuationsrdquo Transactions of the Ameri-can Mathematical Society vol 357 no 10 pp 4191ndash4213 2005

[24] M Ludwig and M Reitzner ldquoA classification of 119878119871(119899) invariantvaluationsrdquoAnnals ofMathematics vol 172 no 2 pp 1219ndash12672010

[25] E Lutwak ldquoThe Brunn-Minkowski-Firey theory I Mixedvolumes and the Minkowski problemrdquo Journal of DifferentialGeometry vol 38 no 1 pp 131ndash150 1993

[26] E Lutwak ldquoThe Brunn-Minkowski-Firey theory II Affine andgeominimal surface areasrdquo Advances in Mathematics vol 118no 2 pp 244ndash294 1996

[27] E Lutwak D Yang and G Zhang ldquo119871119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 56 no 1 pp111ndash132 2000

[28] E Lutwak D Yang and G Zhang ldquoA new ellipsoid associatedwith convex bodiesrdquoDukeMathematical Journal vol 104 no 3pp 375ndash390 2000

[29] E Lutwak D Yang and G Zhang ldquoOn the L119901-Minkowskiproblemrdquo Transactions of the American Mathematical Societyvol 356 no 11 pp 4359ndash4370 2004

[30] E Lutwak D Yang and G Zhang ldquo119871119901 John ellipsoidsrdquoProceedings of the London Mathematical Society vol 90 no 2pp 497ndash520 2005

[31] E Lutwak D Yang and G Zhang ldquoOptimal Sobolev normsand the 119871

119901 Minkowski problemrdquo International MathematicsResearch Notices vol 2006 21 pages 2006

[32] EWerner andD Ye ldquoNew 119871119901 affine isoperimetric inequalitiesrdquoAdvances in Mathematics vol 218 no 3 pp 762ndash780 2008

[33] E Werner and D Ye ldquoInequalities for mixed 119901-affine surfaceareardquoMathematische Annalen vol 347 no 3 pp 703ndash737 2010

[34] H Federer Geometric Measure Theory Springer Berlin Ger-many 1969

[35] T Wang ldquoThe affine Polya-Szego principle equality cases andstabilityrdquo Journal of Functional Analysis vol 265 no 8 pp 1728ndash1748 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

Proof It was shown in [4 (63)] that the following differentialinequality holds

119881(⟨119891⟩119905)(119899minus119901)119899

ge 119899119901minus1

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

(41)

Integrating both sides of the inequality gives

int

infin

0

119881(⟨119891⟩119905)(119899minus119901)119899

119889119905

ge 119899119901minus1

int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(42)

Noting that ℎ119870(sdot) = sdot 119870119900 and Combined with (5) we obtainthat

intR119899

ℎ119870(nabla119891)119901119889119909

ge 119899119901119881(119870)119901119899

int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(43)

By the homogeneous of 119870 in (43) and (40) we only need toconsider 119881(119870) = 120581119899 So it is sufficient to prove that

intR119899

ℎ119870(nabla119891119870)119901

119889119909 = 119899119901120581119901119899

119899int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(44)

The last equality is shown in [3]Nowwe prove this equal-ity by using Lemma 2 Together with the co-area formula (21)the equality (24) the definition of 119888(119891 119905) in Lemma 2 (13)and 119881(119870) = 120581119899 we obtain

intR119899

ℎ119870(nabla119891119870(119909))119901

119889119909

= int

infin

0

(int120597[119891119870]

119905

ℎ119870(nabla119891119870(119909))119901

1003816100381610038161003816nabla119891119870 (119909)

1003816100381610038161003816

119889H119899minus1

(119909))119889119905

= int

infin

0

int120597[119891119870]

119905

ℎ119870(] (119909))11990110038161003816100381610038161003816nabla11989111987010038161003816100381610038161003816

119901minus1

119889H119899minus1

(119909) 119889119905

= int

infin

0

int119878119899minus1

ℎ119870(119906)119901119889119878119901 (⟨119891

119870⟩119905 119906) 119889119905

= int

infin

0

int119878119899minus1

119888 (119891 119905) ℎ119870(119906)119901119889119878119901 (119870 119906) 119889119905

= int

infin

0

119888 (119891 119905) 119889119905 int119878119899minus1

ℎ119870(119906)119901119889119878119901 (119870 119906)

= 119899119881 (119870)int

infin

0

119899119901minus1

120581(119901minus119899)119899

119899120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

= 119899119901120581119901119899

119899int

infin

0

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

119889119905

(45)

where ](119909) = minusnabla119891119870(119909)|nabla119891

119870(119909)| on 120597[119891

119870]119905for almost every

119905 gt 0 And the second equality holds since 119870 is origin-symmetric and the support function of119870 is homogeneous ofdegree 1

MoreoverTheorem 5 can be proved for119901 ge 1 by using thesolution to the even normalized 119871119901Minkowski problem as in[7 9] More precisely suppose 119891 isin 119862

infin

0(R119899) for 119901 ge 1 and

define the normalized 119871119901 convexification ⟨119891⟩

119905as the unique

origin-symmetric convex body such that

1

119881 (⟨119891⟩119905)

int119878119899minus1

119892 (119906) 119889119878119901 (⟨119891⟩119905 119906)

= int120597[119891]119905

119892 (] (119909)) 1003816100381610038161003816nabla1198911003816100381610038161003816119901minus1

119889H119899minus1

(119909)

(46)

for almost every 119905 gt 0 By taking slight modifications in theproof of Theorem 1 we obtain

intR119899

ℎ119870(nabla119891)119901119889119909 ge 119899119881(119870)

119901119899int

infin

0

119881(⟨119891⟩119905)minus119901119899

119889119905 (47)

Similar to the proof ofTheorem 5 together with the observa-tion in [7 (422)] that

119881(⟨119891⟩119905)minus119901119899

ge 119899119901minus1

120583119891(119905)(119899minus1)119901119899

(minus1205831015840

119891(119905))1minus119901

(48)

we also get (43) SoTheorem 5 remains true for 119901 = 119899

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

All the authors contributed equally to the paper All theauthors read and approved the final paper

Acknowledgments

This work is supported in part by the National NaturalScience Foundation of China (Grant no 11371239) ShanghaiLeading Academic Discipline Project (Project no J50101)and the Research Fund for the Doctoral Programs of HigherEducation of China (20123108110001)

References

[1] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951

[2] G Talenti ldquoBest constant in Sobolev inequalityrdquo Annali diMatematica Pura ed Applicata Serie Quarta vol 110 pp 353ndash372 1976

[3] A Alvino V Ferone and G Trombetti ldquoConvex symmetriza-tion and applicationsrdquo Annales de lrsquoInstitut Henri Poincare CNon Linear Analysis vol 14 no 2 pp 275ndash293 1997

[4] E Lutwak D Yang and G Zhang ldquoSharp affine 119871119901 Sobolevinequalitiesrdquo Journal of Differential Geometry vol 62 no 1 pp17ndash38 2002

[5] G Zhang ldquoThe affine Sobolev inequalityrdquo Journal of DifferentialGeometry vol 53 no 1 pp 183ndash202 1999

6 The Scientific World Journal

[6] D Alonso-Gutierrez J Bastero and J Bernues ldquoFactoringSobolev inequalities through classes of functionsrdquo Proceedingsof the American Mathematical Society vol 140 no 10 pp 3557ndash3566 2012

[7] A Cianchi E Lutwak D Yang and G Zhang ldquoAffine Moser-Trudinger and MORrey-Sobolev inequalitiesrdquo Calculus of Vari-ations and Partial Differential Equations vol 36 no 3 pp 419ndash436 2009

[8] C Haberl and F E Schuster ldquoAsymmetric affine 119871119901 Sobolevinequalitiesrdquo Journal of Functional Analysis vol 257 no 3 pp641ndash658 2009

[9] C Haberl F Schuster and J Xiao ldquoAn asymmetric affine Polya-Szego principlerdquo Mathematische Annalen vol 352 no 3 pp517ndash542 2012

[10] J E Brothers and W P Ziemer ldquoMinimal rearrangementsof Sobolev functionsrdquo Journal fur die Reine und AngewandteMathematik vol 384 pp 153ndash179 1988

[11] A Cianchi L Esposito N Fusco and T Trombetti ldquoA quanti-tative Polya-Szego principlerdquo Journal fur die Reine und Ange-wandte Mathematik vol 614 pp 153ndash189 2008

[12] A Cianchi and N Fusco ldquoFunctions of bounded variation andrearrangementsrdquo Archive for Rational Mechanics and Analysisvol 165 no 1 pp 1ndash40 2002

[13] L Esposito and P Ronca ldquoQuantitative Polya-Szego principlefor convex symmetrizationrdquo Manuscripta Mathematica vol130 no 4 pp 433ndash459 2009

[14] L Esposito and C Trombetti ldquoConvex symmetrization andPolya-Szego inequalityrdquo Nonlinear Analysis Theory Methods ampApplications vol 56 no 1 pp 43ndash62 2004

[15] A Ferone and R Volpicelli ldquoMinimal rearrangements ofSobolev functions a new proofrdquo Annales de lrsquoInstitut HenriPoincare (C) Non Linear Analysis vol 20 no 2 pp 333ndash3392003

[16] A Ferone and R Volpicelli ldquoConvex rearrangement equalitycases in the Polya-Szego inequalityrdquo Calculus of Variations andPartial Differential Equations vol 21 no 3 pp 259ndash272 2004

[17] M Ludwig J Xiao and G Zhang ldquoSharp convex Lorentz-Sobolev inequalitiesrdquo Mathematische Annalen vol 350 no 1pp 169ndash197 2011

[18] S Campi and P Gronchi ldquoThe 119871119901-Busemann-Petty centroid

inequalityrdquoAdvances in Mathematics vol 167 no 1 pp 128ndash1412002

[19] S Campi and P Gronchi ldquoOn the reverse 119871119901-Busemann-Pettycentroid inequalityrdquo Mathematika vol 49 no 1-2 pp 1ndash11(2004) 2002

[20] C Haberl and F E Schuster ldquoGeneral L119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 83 no 1 pp1ndash26 2009

[21] D Hug E Lutwak D Yang and G Zhang ldquoOn the 119871119901 Minko-wski problem for polytopesrdquo Discrete amp Computational Geom-etry vol 33 no 4 pp 699ndash715 2005

[22] M Ludwig ldquoEllipsoids and matrix-valued valuationsrdquo DukeMathematical Journal vol 119 no 1 pp 159ndash188 2003

[23] M Ludwig ldquoMinkowski valuationsrdquo Transactions of the Ameri-can Mathematical Society vol 357 no 10 pp 4191ndash4213 2005

[24] M Ludwig and M Reitzner ldquoA classification of 119878119871(119899) invariantvaluationsrdquoAnnals ofMathematics vol 172 no 2 pp 1219ndash12672010

[25] E Lutwak ldquoThe Brunn-Minkowski-Firey theory I Mixedvolumes and the Minkowski problemrdquo Journal of DifferentialGeometry vol 38 no 1 pp 131ndash150 1993

[26] E Lutwak ldquoThe Brunn-Minkowski-Firey theory II Affine andgeominimal surface areasrdquo Advances in Mathematics vol 118no 2 pp 244ndash294 1996

[27] E Lutwak D Yang and G Zhang ldquo119871119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 56 no 1 pp111ndash132 2000

[28] E Lutwak D Yang and G Zhang ldquoA new ellipsoid associatedwith convex bodiesrdquoDukeMathematical Journal vol 104 no 3pp 375ndash390 2000

[29] E Lutwak D Yang and G Zhang ldquoOn the L119901-Minkowskiproblemrdquo Transactions of the American Mathematical Societyvol 356 no 11 pp 4359ndash4370 2004

[30] E Lutwak D Yang and G Zhang ldquo119871119901 John ellipsoidsrdquoProceedings of the London Mathematical Society vol 90 no 2pp 497ndash520 2005

[31] E Lutwak D Yang and G Zhang ldquoOptimal Sobolev normsand the 119871

119901 Minkowski problemrdquo International MathematicsResearch Notices vol 2006 21 pages 2006

[32] EWerner andD Ye ldquoNew 119871119901 affine isoperimetric inequalitiesrdquoAdvances in Mathematics vol 218 no 3 pp 762ndash780 2008

[33] E Werner and D Ye ldquoInequalities for mixed 119901-affine surfaceareardquoMathematische Annalen vol 347 no 3 pp 703ndash737 2010

[34] H Federer Geometric Measure Theory Springer Berlin Ger-many 1969

[35] T Wang ldquoThe affine Polya-Szego principle equality cases andstabilityrdquo Journal of Functional Analysis vol 265 no 8 pp 1728ndash1748 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

[6] D Alonso-Gutierrez J Bastero and J Bernues ldquoFactoringSobolev inequalities through classes of functionsrdquo Proceedingsof the American Mathematical Society vol 140 no 10 pp 3557ndash3566 2012

[7] A Cianchi E Lutwak D Yang and G Zhang ldquoAffine Moser-Trudinger and MORrey-Sobolev inequalitiesrdquo Calculus of Vari-ations and Partial Differential Equations vol 36 no 3 pp 419ndash436 2009

[8] C Haberl and F E Schuster ldquoAsymmetric affine 119871119901 Sobolevinequalitiesrdquo Journal of Functional Analysis vol 257 no 3 pp641ndash658 2009

[9] C Haberl F Schuster and J Xiao ldquoAn asymmetric affine Polya-Szego principlerdquo Mathematische Annalen vol 352 no 3 pp517ndash542 2012

[10] J E Brothers and W P Ziemer ldquoMinimal rearrangementsof Sobolev functionsrdquo Journal fur die Reine und AngewandteMathematik vol 384 pp 153ndash179 1988

[11] A Cianchi L Esposito N Fusco and T Trombetti ldquoA quanti-tative Polya-Szego principlerdquo Journal fur die Reine und Ange-wandte Mathematik vol 614 pp 153ndash189 2008

[12] A Cianchi and N Fusco ldquoFunctions of bounded variation andrearrangementsrdquo Archive for Rational Mechanics and Analysisvol 165 no 1 pp 1ndash40 2002

[13] L Esposito and P Ronca ldquoQuantitative Polya-Szego principlefor convex symmetrizationrdquo Manuscripta Mathematica vol130 no 4 pp 433ndash459 2009

[14] L Esposito and C Trombetti ldquoConvex symmetrization andPolya-Szego inequalityrdquo Nonlinear Analysis Theory Methods ampApplications vol 56 no 1 pp 43ndash62 2004

[15] A Ferone and R Volpicelli ldquoMinimal rearrangements ofSobolev functions a new proofrdquo Annales de lrsquoInstitut HenriPoincare (C) Non Linear Analysis vol 20 no 2 pp 333ndash3392003

[16] A Ferone and R Volpicelli ldquoConvex rearrangement equalitycases in the Polya-Szego inequalityrdquo Calculus of Variations andPartial Differential Equations vol 21 no 3 pp 259ndash272 2004

[17] M Ludwig J Xiao and G Zhang ldquoSharp convex Lorentz-Sobolev inequalitiesrdquo Mathematische Annalen vol 350 no 1pp 169ndash197 2011

[18] S Campi and P Gronchi ldquoThe 119871119901-Busemann-Petty centroid

inequalityrdquoAdvances in Mathematics vol 167 no 1 pp 128ndash1412002

[19] S Campi and P Gronchi ldquoOn the reverse 119871119901-Busemann-Pettycentroid inequalityrdquo Mathematika vol 49 no 1-2 pp 1ndash11(2004) 2002

[20] C Haberl and F E Schuster ldquoGeneral L119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 83 no 1 pp1ndash26 2009

[21] D Hug E Lutwak D Yang and G Zhang ldquoOn the 119871119901 Minko-wski problem for polytopesrdquo Discrete amp Computational Geom-etry vol 33 no 4 pp 699ndash715 2005

[22] M Ludwig ldquoEllipsoids and matrix-valued valuationsrdquo DukeMathematical Journal vol 119 no 1 pp 159ndash188 2003

[23] M Ludwig ldquoMinkowski valuationsrdquo Transactions of the Ameri-can Mathematical Society vol 357 no 10 pp 4191ndash4213 2005

[24] M Ludwig and M Reitzner ldquoA classification of 119878119871(119899) invariantvaluationsrdquoAnnals ofMathematics vol 172 no 2 pp 1219ndash12672010

[25] E Lutwak ldquoThe Brunn-Minkowski-Firey theory I Mixedvolumes and the Minkowski problemrdquo Journal of DifferentialGeometry vol 38 no 1 pp 131ndash150 1993

[26] E Lutwak ldquoThe Brunn-Minkowski-Firey theory II Affine andgeominimal surface areasrdquo Advances in Mathematics vol 118no 2 pp 244ndash294 1996

[27] E Lutwak D Yang and G Zhang ldquo119871119901 affine isoperimetricinequalitiesrdquo Journal of Differential Geometry vol 56 no 1 pp111ndash132 2000

[28] E Lutwak D Yang and G Zhang ldquoA new ellipsoid associatedwith convex bodiesrdquoDukeMathematical Journal vol 104 no 3pp 375ndash390 2000

[29] E Lutwak D Yang and G Zhang ldquoOn the L119901-Minkowskiproblemrdquo Transactions of the American Mathematical Societyvol 356 no 11 pp 4359ndash4370 2004

[30] E Lutwak D Yang and G Zhang ldquo119871119901 John ellipsoidsrdquoProceedings of the London Mathematical Society vol 90 no 2pp 497ndash520 2005

[31] E Lutwak D Yang and G Zhang ldquoOptimal Sobolev normsand the 119871

119901 Minkowski problemrdquo International MathematicsResearch Notices vol 2006 21 pages 2006

[32] EWerner andD Ye ldquoNew 119871119901 affine isoperimetric inequalitiesrdquoAdvances in Mathematics vol 218 no 3 pp 762ndash780 2008

[33] E Werner and D Ye ldquoInequalities for mixed 119901-affine surfaceareardquoMathematische Annalen vol 347 no 3 pp 703ndash737 2010

[34] H Federer Geometric Measure Theory Springer Berlin Ger-many 1969

[35] T Wang ldquoThe affine Polya-Szego principle equality cases andstabilityrdquo Journal of Functional Analysis vol 265 no 8 pp 1728ndash1748 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of