Problems related to the Kneser-Poulsen conjecture

Maria Belk

A Bunch of Discs

A Bunch of Discs

Suppose we rearrange the discs so that the distances between centers increases.

A Bunch of Discs

Suppose we rearrange the discs so that the distances between centers increases.

A Bunch of Discs

What do you think happens to the area of the union of the discs?

A Bunch of Discs

What do you think happens to the area of the union of the discs? Probably increases.

A Bunch of Discs

What do you think happens to the area of the intersection?

A Bunch of Discs

What do you think happens to the area of the intersection?

A Bunch of Discs

What do you think happens to the area of the intersection?

A Bunch of Discs

What do you think happens to the area of the intersection? Probably decreases.

Kneser-Poulsen Conjecture

Conjecture (Kneser 1955, Poulsen 1954) If the distances between the centers of the discs have not decreased, then the area of the union has either increased or remained the same.

Also conjectured: The area of the intersection has either decreased or remained the same.

Kneser-Poulsen Theorem

Theorem (Bezdek and Connelly, 2002)If the distances between the centers of the discs have not decreased, then:

• The area of the union has either decreased or remained the same.

• The area of the intersection has either increased or remained the same.

Kneser-Poulsen in other spaces?

We can ask the analogous question in:• Higher dimensions , • Spherical Space

• Hyperbolic Space

Kneser-Poulsen in other spaces?

We can ask the analogous question in:• Higher dimensions , • Spherical Space , • Hyperbolic Space

Kneser-Poulsen in other spaces?

We can ask the analogous question in:• Higher dimensions , • Spherical Space , • Hyperbolic Space ,

Notation

• p = (, , , , ) is a configuration of points in .

Notation

• p = (, , , , ) is a configuration of points in .

• q is also a configuration of points in .

Notation

We say that q is an expansion of p, if

, ,

Notation

We say that p continuously expands to q if there is a continuous motion from p to q, in which the distances change monotonically.

Notation

We say that p continuously expands to q if there is a continuous motion from p to q, in which the distances change monotonically.

Notation

We say that p continuously expands to q if there is a continuous motion from p to q, in which the distances change monotonically.

Continuous Case

In , , and , Csikós has shown:

Theorem (Csikós 1999, 2002) If there is a continuous expansion between the two configurations, then the volume behaves appropriately.

Wall

Why? Because = , where

= size of Wall between discs and

= change in distance between and

Continuous Case

Wall

Why? Because = , where

= size of Wall between discs and

= change in distance between and

Continuous Case

Why? Because = , where

= size of Wall between discs and

= change in distance between and

Continuous Case

For more than 2 balls:

=

The walls come from the Voronoi diagram.

Walls

Kneser-Poulsen in other spaces?

Theorem (Csikós, 2006) Euclidean, Hyperbolic, and Spherical spaces are the only reasonable spaces to ask the question in.

Counterexamples exist if the space is• Not homogeneous• Not isotropic• Not simply connected

On a Cylinder:

An expansion, where the area of the union decreases:

On a Cylinder:

An expansion, where the area of the union decreases:

What is known?

In Euclidean space:• Gromov: balls in dimensions.• Bern and Sahai: Discs in two dimensions if

there is a continuous expansion.• Csikós: Any dimension if there is a continuous

expansion.• Bezdek and Connelly: Discs in two dimensions

(no continuous expansion needed).

What is known?

Spherical and Hyperbolic Space:• Csikós: Any dimension if there is a continuous

expansion.

Outline

• Sketch of proof for dimension 2.

• The problems in extending this proof to higher dimensions?

• Hyperbolic and spherical spaces?

• Tensegrities in Hyperbolic space

• Remaining Questions

Outline

• Sketch of proof for dimension 2.

• The problems in extending this proof to higher dimensions?

• Hyperbolic and spherical spaces?

• Tensegrities in Hyperbolic space

• Remaining Questions

Outline

• Sketch of proof for dimension 2.

• The problems in extending this proof to higher dimensions?

• Hyperbolic and spherical spaces?

• Tensegrities in Hyperbolic space

• Remaining Questions

Outline

• Sketch of proof for dimension 2.

• The problems in extending this proof to higher dimensions?

• Hyperbolic and spherical spaces?

• Tensegrities in Hyperbolic space

• Remaining Questions

Outline

• Sketch of proof for dimension 2.

• The problems in extending this proof to higher dimensions?

• Hyperbolic and spherical spaces?

• Tensegrities in Hyperbolic space

• Remaining Questions

Proof of Kneser-Poulsen:

The proof due to Bezdek and Connelly has two main components:

1. Lemma: If there is a continuous expansion from p to q in dim , then volume in dim does not decrease.

2. Lemma: If q is an expansion of p, in dim , then there is a continuous expansion in .

Since , the conjecture holds.

Lemma 1

Lemma (Bezdek and Connelly): If there is a continuous expansion from p to q in dim , then volume in dim does not decrease.

Idea of Proof: Use Cylindrical Shells to relate the volume in dimension to the surface area in dimension .

Lemma 1

Lemma (Bezdek and Connelly): If there is a continuous expansion from p to q in dim , then volume in dim does not decrease.

Sketch of Proof:Discs (in 1 dimension, equal radii, for simplicity)

Lemma (Bezdek and Connelly): If there is a continuous expansion from p to q in dim , then volume in dim does not decrease.

1.Create Voronoi Diagram

Discs (in 1 dimension, equal radii, for simplicity)Sketch of Proof:

Lemma (Bezdek and Connelly): If there is a continuous expansion from p to q in dim , then volume in dim does not decrease.

1.Create Voronoi Diagram

Discs (in 1 dimension, equal radii, for simplicity)Sketch of Proof:

Lemma (Bezdek and Connelly): If there is a continuous expansion from p to q in dim , then volume in dim does not decrease.

1.Create Voronoi Diagram2.Create balls in 3 dimensions

Discs (in 1 dimension, equal radii, for simplicity)Sketch of Proof:

Lemma (Bezdek and Connelly): If there is a continuous expansion from p to q in dim , then volume in dim does not decrease.

1. Create Voronoi Diagram2. Create balls in 3 dimensions.3. Consider cylindrical shells.

Discs (in 1 dimension, for simplicity)Sketch of Proof:

Sketch of Proof

Result of Cylindrical Shells for each Voronoi region separately:

Sketch of Proof

• Differentiate to get: Vol. in dim. = Surface area in dim

• If there is a continuous motion in dim , then the surface area changes monotonically between the 2 configurations.

• Thus, the volume in dim does not change.

Lemma 2

Lemma (well-known): If q is an expansion of p, in dim , then there is a continuous expansion in .

Proof: Place p and q in orthogonal subspaces, then the following motion works:

The problems of extending this proof to higher dimensions.

Recall

Lemma (Bezdek and Connelly) If there is a continuous expansion from p to q in dim , then volume in dim does not decrease.

Question

Lemma (Bezdek and Connelly) If there is a continuous expansion from p to q in dim , then volume in dim does not decrease.

Question: If p is an expansion of q in , in what dimension is there a continuous expansion?

4

1

4

3

2

1 3

2

p q

ExampleThis expansion in 2 dimensions requires 3

dimensions for a continuous expansion.

4

1

4

3

2

1 3

2

p q

Another Example

Another ExampleContinuous contraction in dimension 4.

In Dimensions

An analogous construction in :

1. Start with Tetrahedron:

In Dimensions

An analogous construction in :

1. Start with Tetrahedron.

2. Attach “flaps” to each face.

In Dimensions

An analogous construction in :

1. Start with Tetrahedron.

2. Attach “flaps” to each face.

The result requires 6 dimensionsto continuously contract.

In dimensions

Analogous construction in :

1. Start with a -simplex.

2. Attach “flaps” to each facet.

Theorem (Belk and Connelly) Requires dimensions to continuously contract, because the bar framework is rigid in .

What does this mean?

Question: If q is an expansion of p in , in what dimension is there a continuous expansion?

Answer: There is a continuous expansion in , and we cannot do better than that.

If we wanted to use a similar proof for higher dimensions, we would need to improve the other lemma.

Hyperbolic and Spherical Spaces

Hyperbolic and Spherical Space

Theorem (Csikós, 2002) If there is a continuous expansion in (or ), then Kneser-Poulsen holds in (or ).

Question: If q is an expansion of p in or , in what dimension is there a continuous expansion?

Spherical Space

Question: If q is an expansion of p in , in what dimension is there a continuous expansion?

Since p and q sit in , there is a continuous expansion in , which remains on the sphere .

Therefore: There is a continuous expansion in .

Spherical Space

This is not enough to prove Kneser-Poulsen in , but we do get:

Theorem (Csikós 2002) Kneser-Poulsen holds for balls in .

Since balls continuously expand in .

Hyperbolic Space

Question: If q is an expansion of p in , is there a continuous expansion from p to q in for any ?

This is unknown.

Tensegrities

Tensegrity

A tensegrity p is a configuration p and a graph where each edge of the graph is labeled as a cable, strut, or bar.

Cable Can become shorter

Strut Can become longer

Bar Must remain the same length

Tensegrity

Example:

This tensegrity can flex — the vertices can be moved while maintaining the cable/strut conditions.

Tensegrity

Example:

We can stretch the strut until the points are collinear.

Tensegrity

Example:

Now, the tensegrity is rigid — the vertices cannot be moved while maintaining the cable/strut conditions.

Tensegrity

A more complicated example:

This tensegrity is also rigid.

Tensegrity

Another example:

This tensegrity is rigid in , but it flexes in .

What tensegrities are we interested in?

Suppose q is an expansion of p. Create the tensegrity:

1. The configuration is p.

2. If the distance between two vertices increases from p to q, make the edge between the vertices a strut.

3. If the distance remains the same, make the edge a bar.

What tensegrities are we interested in?

This creates tensegrities where:

1. is the complete graph (that is, there is an edge between any two vertices).

2. Every edge is either a bar or strut.

3. There exists another configuration q, which satisfies the bar and strut conditions. This means the tensegrity is not globally rigid.

Global Rigidity

Global Rigidity: A tensegrity is globally rigid if there is no other configuration satisfying the cable, strut, and bar conditions.

GloballyRigid

GloballyRigid

Not GloballyRigid

In Euclidean Space

In , for large enough , every tensegrity is either:• Globally rigid, or• Flexible

Because: If it is not globally rigid, there is a motion connecting the two configurations.

In Hyperbolic Space

Open Problem: In (for large enough ), is every tensegrity either globally rigid or flexible?

Lemma (Belk) The following tensegrities are either globally rigid or flexible in .

1. Tensegrities with fewer than 4 points, and

2. Tensegrities with points in general position that span .

Why? • Case Checking• Minimal Tensegrities (tensegrities in general

position in dimensions)• Pogorelov Map (:

, complicated

function with some nice properties)

In Hyperbolic Space

First tensegrity for which it is not known:

Globally rigid in ?(It is globally rigid in .)

Minimal Tensegrities

1. Start with 2 simplices with exactly one point in common.

or

Minimal Tensegrities

1. Start with 2 simplices with exactly one point in common.

2. Replace the edges of simplices with struts.

or

Minimal Tensegrities

1. Start with 2 simplices with exactly one point in common.

2. Replace the edges of simplices with struts.

3. Add cables between all remaining vertices.

or

Minimal Tensegrities

These are the only rigid tensegrities in dim with vertices in general position.

They are globally rigid in and .

or

Pogorelov Map

p, q (r, s)

p, q = configurations in

r, s = configurations in

With the property that:

, , , ,

Pogorelov Map

The problem is that the Pogorelov Map can significantly change the configuration.

Hyperbolic Space Euclidean Space

Pogorelov Map

We can fix this problem for minimal tensegrities, by specifying where the point of intersection of the simplicies goes in Euclidean space..

Hyperbolic Space Euclidean Space

Result

Theorem (Belk): Kneser-Poulsen holds for 4 balls in (for any ).

Remaining Questions

Euclidean Space

Euclidean Space:

1. Kneser-Poulsen in ?2. Could the simplex with “flaps” provide a

counter-example?

Spherical

Spherical Space:

1. Kneser-Poulsen in ?2. If q is an expansion of p in , is there a

continuous expansion in ?

Hyperbolic Space

Hyperbolic Space:

1. Kneser-Poulsen in ?2. If q is an expansion of p in , is there a

continuous expansion in (for any )?3. Is global rigidity for tensegrities equivalent in

Hyperbolic space and Euclidean space?