From r-dual sets to uniform contractions

Abstract

Let $$\mathbb {M}^d$$ denote the d-dimensional Euclidean, hyperbolic, or spherical space. The r-dual set of a given set in $$\mathbb {M}^d$$ is the intersection of balls of radii r centered at the points of the a given set. In this paper we prove that for any set of given volume in $$\mathbb {M}^d$$ the volume of the r-dual set becomes maximal if the set is a ball. As an application we prove the following. The Kneser–Poulsen Conjecture states that if the centers of a family of N congruent balls in Euclidean d-space is contracted, then the volume of the intersection does not decrease. A uniform contraction is a contraction 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. We prove a special case of the Kneser–Poulsen conjecture namely, we prove the conjecture for uniform contractions (with sufficiently large N) in $$\mathbb {M}^d$$ .