Problems related to the Kneser-Poulsen conjecture

1. Problems related to the Kneser-Poulsen conjecture Maria Belk

2. A Bunch of Discs

3. A Bunch of Discs Suppose we rearrange the discs so that the distances between centers increases.

4. A Bunch of Discs Suppose we rearrange the discs so that the distances between centers increases.

5. A Bunch of Discs What do you think happens to the area of the union of the discs?

6. A Bunch of Discs What do you think happens to the area of the union of the discs? Probably increases.

7. A Bunch of Discs What do you think happens to the area of the intersection?

8. A Bunch of Discs What do you think happens to the area of the intersection?

9. A Bunch of Discs What do you think happens to the area of the intersection?

10. A Bunch of Discs What do you think happens to the area of the intersection? Probably decreases.

11. 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.

12. 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.

13. Kneser-Poulsen in other spaces? We can ask the analogous question in: • Higher dimensions ,  • Spherical Space  • Hyperbolic Space 

14. Kneser-Poulsen in other spaces? We can ask the analogous question in: • Higher dimensions ,  • Spherical Space ,  • Hyperbolic Space 

15. Kneser-Poulsen in other spaces? We can ask the analogous question in: • Higher dimensions ,  • Spherical Space ,  • Hyperbolic Space , 

16.      Notation • p = (, , , , ) is a configuration of points in .

17.      Notation • p = (, , , , ) is a configuration of points in . • q is also a configuration of  points in .

18. Notation We say that q is an expansion of p, if ,   ,           

19. Notation We say that pcontinuously expands to q if there is a continuous motion from p to q, in whichthe distances change monotonically.

20. Notation We say that pcontinuously expands to q if there is a continuous motion from p to q, in whichthe distances change monotonically.

21. Notation We say that pcontinuously expands to q if there is a continuous motion from p to q, in whichthe distances change monotonically.

22. Wall 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. Why? Because  =  , where  = size of Wall between discs  and   = change in distance between  and 

23. Continuous Case   Wall Why? Because  =  , where  = size of Wall between discs  and   = change in distance between  and 

24. Continuous Case   Why? Because  =  , where  = size of Wall between discs  and   = change in distance between  and 

25. Continuous Case For more than 2 balls: Walls  =   The walls come from the Voronoi diagram.

26. 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

27. On a Cylinder: An expansion, where the area of the union decreases:

28. On a Cylinder: An expansion, where the area of the union decreases:

29. 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).

30. What is known? Spherical and Hyperbolic Space: • Csikós: Any dimension if there is a continuous expansion.

31. 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

32. 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

33. 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

34. 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

35. 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

36. Proof of Kneser-Poulsen: The proof due to Bezdek and Connelly has two main components: • Lemma: If there is a continuous expansion from p to q in dim , then volume in dim  does not decrease. • Lemma: If q is an expansion of p, in dim , then there is a continuous expansion in . Since       , the conjecture holds.

37. 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 .

38. 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)

39. 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) • Create Voronoi Diagram

40. 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) • Create Voronoi Diagram

41. 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) • Create Voronoi Diagram • Create balls in 3 dimensions

42. 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, for simplicity) • Create Voronoi Diagram • Create balls in 3 dimensions. • Consider cylindrical shells.

43. Sketch of Proof Result of Cylindrical Shells for each Voronoi region separately:

44. 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.

45. 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:

46. The problems of extending this proof to higher dimensions.

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

48. 2 4 3 1 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 2 3 1 q p

49. 2 4 3 1 Example This expansion in 2 dimensions requires 3 dimensions for a continuous expansion. 4 2 3 1 q p

50. Another Example