The Kneser--Poulsen conjecture for special contractions

TL;DR

This paper proves the Kneser--Poulsen conjecture for special types of contractions, specifically uniform and strong contractions, showing volume non-increase or non-decrease under these conditions.

Contribution

The paper introduces and proves the conjecture for uniform and strong contractions, extending results to intrinsic volumes and unconditional bodies.

Findings

Uniform contractions do not decrease the volume of intersections for sufficiently many balls.

The conjecture holds for strong contractions and extends to unconditional bodies.

Results apply to intrinsic volumes beyond volume.

Abstract

The Kneser--Poulsen Conjecture states that if the centers of a family of $N$ unit balls in ${\mathbb E}^d$ is contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions. First, a \emph{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 obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that $N\geq(1+\sqrt{2})^d$. Our result extends to intrinsic volumes. We prove a similar result concerning the volume of the union. Second, a \emph{strong contraction} is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls.