From $r$-dual sets to uniform contractions

Let $M^d$ denote the $d$-dimensional Euclidean, hyperbolic, or spherical space. The $r$-dual set of given set in $M^d$ is the intersection of balls of radii $r$ centered at the points of the given set. In this paper we prove that for any set of given volume in $M^d$ 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. We prove the Kneser-Poulsen conjecture for uniform contractions (with $N$ sufficiently large) in $M^d$.