# On uniform contractions of balls in Minkowski spaces

## TL;DR

This paper investigates how uniform contractions of ball centers in Minkowski spaces affect the volume of their unions and intersections, establishing volume invariance under certain conditions and providing improvements for Euclidean spaces.

## Contribution

It proves that uniform contractions do not increase union volume or decrease intersection volume of balls in Minkowski spaces under specific conditions, extending and improving known results.

## Key findings

- Volume of union does not increase under uniform contraction for N ≥ 2^d.
- Volume of intersection does not decrease under uniform contraction for N ≥ 3^d.
- Improved results are provided specifically for Euclidean spaces.

## Abstract

Let $N$ balls of the same radius be given in a $d$-dimensional real normed vector space, i.e., in a Minkowski $d$-space. Then apply a uniform contraction to the centers of the $N$ balls without changing the common radius. Here 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. The main results of this paper state that a uniform contraction of the centers does not increase (resp., decrease) the volume of the union (resp., intersection) of $N$ balls in Minkowski $d$-space, provided that $N\geq 2^d$ (resp., $N\geq 3^d$ and the unit ball of the Minkowski $d$-space is a generating set). Some improvements are presented in Euclidean spaces.