On the volume of unions and intersections of congruent balls under uniform contractions

Károly Bezdek

Published 2019-03-07Version 1

Let ${\mathbb E}^d$ denote the $d$-dimensional Euclidean space. According to the longstanding Kneser-Poulsen conjecture (resp., Gromov-Klee-Wagon problem) if the centers of a family of $N$ congruent balls in ${\mathbb E}^d$, $d>2$ are contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). In this note, we investigate these problems for uniform contractions, which are contractions where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers, that is, when the pairwise distances of the two sets are separated by some positive real number. Improving earlier results, we prove that the volume of the union of $N$ congruent balls in ${\mathbb E}^d$, $d>2$ does not increase under any uniform contraction of the center points when $ N\geq 2^d$. Furthermore, we show that the volume of the intersection of $N$ congruent balls in ${\mathbb E}^d$ does not decrease under any uniform contraction of the center points when $ N\geq {2.359}^d$ and $d\geq d_0$, where $d_0$ is a (large) universal constant.