The Central Set and Its Application to the Kneser–Poulsen Conjecture

Abstract

The Kneser-Poulsen conjecture says that if a finite collection of balls in a Euclidean (spherical or hyperbolic) space is rearranged so that the distance between each pair of centers does not increase, then the volume of the union of these balls does not increase as well. We give new results about central sets of subsets of a Riemannian manifold and apply these results to prove new special cases of the Kneser-Poulsen conjecture in the twodimensional sphere and the hyperbolic plane.51M16 (51M25, 51M10, 53C20, 53C22) arXiv:1511.08134v3 [math.MG]

Two Kneser–Poulsen-type inequalities in planes of constant curvature

“…, p k−1 , and Y is the segment [p k−1 , p k ]. Following [4], for a closed subset A ⊆ C U , we denote by U A the union of those maximal disks in U , the centers of which belong to A. We prove that in our case,…”

“…The Euclidean analogue of the latter statement was proved by K. Bezdek and R. Connelly [2]. Both theorems are proved by a suitable adaptation of a recently published method of I. Gorbovickis [4].…”

“…As U is convex, hence homotopically trivial, the central set C U is also contractible, so it must be a tree, the edges of which are geodesic segments. For details of the proof, we refer to Section 4 of [4], for illustration, see Figure 4.…”

“…In a recent paper, I. Gorbovickis [4] introduced a new method, with the help of which he could extend the result of K. Bezdek and R. Connelly to the hyperbolic and spherical planes under some extra assumptions. Namely, he proved the following theorems.…”

“…The Euclidean analogue of the latter statement was proved by K. Bezdek and R. Connelly [2]. Both theorems are proved by a suitable adaptation of a recently published method of I. Gorbovickis [4].The following consequence of the Euclidean Kneser-Poulsen conjecture was proved by V. N. Sudakov [8], R. Alexander [1], V. Capoyleas and J. Pach [3].…”