On the volume of the union of balls
type: Item
instance of: scholarly article
title: On the volume of the union of balls
zbMATH Open document ID: 0922.51010
DOI: 10.1007/PL00009395
published in: Discrete \& Computational Geometry
publication date: 1999-10-06T00:00:00Z
review text: A longstanding and exasperating conjecture of Poulsen and Kneser states that if a finite collection of balls in \(\mathbb{R}^d\) is moved so that each pair of centers ends up closer than before, the volume of their union cannot thereby be increased. Bollobás proved in 1968 that the conjecture is true for congruent disks in the plane, if the disks are moved so that the distance between each pair of centers is always nonincreasing; and this has since been generalized to noncongruent disks. The present paper extends these results to balls in higher-dimensional spaces, while retaining the condition that distances must never increase. The author derives a formula for the derivative of the volume as a function of the derivatives of the various distances between centers. This immediately yields a proof for differentiable motions. However, it is possible that, while the distances between centers are well-behaved, the paths of individual centers are much less so. Therefore, the result does not extend trivially to arbitrary continuous motions; it may be necessary, in the author's words, to ``calm down a nonrectifiable `shivering' motion of the centers''. It is shown -- via an excursion into compact Lie groups -- that this can be done, and the main result follows.
Mathematics Subject Classification ID: 51M25
zbMATH DE Number: 1239614
zbMATH Keywords: disks
zbMATH Keywords: balls
zbMATH Keywords: volume
zbMATH Keywords: union
zbMATH Keywords: distance
zbMATH Keywords: monotone decreasing
author: Balázs Csikós
reviewed by: Robert Dawson
MaRDI profile type: Publication
full work available at URL: pl00009395
OpenAlex ID: W2060988067
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description: scientific article; zbMATH DE number 1239614