Strict Kneser-Poulsen conjecture for large radii

TL;DR

This paper proves the Kneser-Poulsen conjecture for large radii in Euclidean space, showing volume and surface volume inequalities hold under certain rearrangements, using tensegrity theory and asymptotic analysis.

Contribution

It establishes the conjecture for large radii, introduces new inequalities, and connects mean width with volume asymptotics, extending prior results with novel methods.

Findings

Volume of unions and intersections increases or decreases under rearrangement for large radii.

Mean width of convex hull increases strictly unless rearrangement is a congruence.

Asymptotic expansion of intersection volume relates to mean width of centers.

Abstract

In this paper we prove the Kneser-Poulsen conjecture for the case of large radii. Namely, if a finite number of points in Euclidean space $E^n$ is rearranged so that the distance between each pair of points does not decrease, then there exists a positive number $r_0$ that depends on the rearrangement of the points, such that if we consider $n$-dimensional balls of radius $r>r_0$ with centers at these points, then the volume of the union (intersection) of the balls before the rearrangement is not less (not greater) than the volume of the union (intersection) after the rearrangement. Moreover, the inequality is strict whenever the new point set is not congruent to the original one. Also under the same conditions we prove a similar result about surface volumes instead of volumes. In order to prove the above mentioned results we use ideas from tensegrity theory to strengthen the theorem of Sudakov, R. Alexander and Capoyleas and Pach, which says that the mean width of the convex hull of a finite number of points does not decrease after an expansive rearrangement of those points. In this paper we show that the mean width increases strictly, unless the expansive rearrangement was a congruence. We also show that if the configuration of centers of the balls is fixed and the volume of the intersection of the balls is considered as a function of the radius $r$, then the second highest term in the asymptotic expansion of this function is equal to $-M_n r^{n-1}$, where $M_n$ is the mean width of the convex hall of the centers. This theorem was conjectured by Balazs Csikos in 2009.