Kneser-Poulsen conjecture for a small number of intersections

TL;DR

This paper proves the Kneser-Poulsen conjecture for configurations where each pairwise intersection involves at most d+1 other balls, under specific intersection conditions.

Contribution

It establishes the conjecture's validity for certain configurations based on intersection complexity, extending previous understanding.

Findings

The conjecture holds when each pairwise intersection involves no more than d+1 other balls.

The result applies to configurations with limited intersection complexity.

Provides a new condition under which the Kneser-Poulsen conjecture is true.

Abstract

The Kneser-Poulsen conjecture says that if a finite collection of balls in a d-dimensional Euclidean space is rearranged so that the distance between each pair of centers does not get smaller, then the volume of the union of these balls also does not get smaller. In this paper we prove that if in the initial configuration the intersection of any two balls has common points with no more than d+1 other balls, then the conjecture holds.