Strict Kneser–Poulsen conjecture for large radii

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 \({\mathbb{E}^n}\) 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 (Dokl. Akad. Nauk SSSR 197:43–45, 1971), Alexander (Trans. Am. Math. Soc., 288(2):661–678, 1985) and Capoyleas and Pach (Discrete and computational geometry. American Mathematical Society, Providence, 1991), 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 \({-M_nr^{n-1}}\) , where M n is the mean width of the convex hall of the centers. This theorem was conjectured by Balázs Csikós in 2009.