Borsuk's partition conjecture
type: Item
instance of: scholarly article
title: Borsuk's partition conjecture
zbMATH Open document ID: 1484.52013
published in: Japanese Journal of Mathematics. 3rd Series
publication date: 2022-01-18T00:00:00Z
review text: This is a survey article on the famous problem of \textit{K. Borsuk} [Fundam. Math. 20, 177--190 (1933; Zbl 0006.42403)] who asked if every set of a given diameter (we tacitly assume that over 0) in the Euclidean \(n\)-dimensional space \(E^n\) can be divided into \(n+1\) parts of smaller diameter. First, this article lists some cases when the problem has a positive solution. For instance for \(n=3\) and for every convex body with smooth boundary in \(E^n\). Then much attention is paid to the result by \textit{J. Kahn} and \textit{G. Kalai} [Bull. Am. Math. Soc., New Ser. 29, No. 1, 60--62 (1993; Zbl 0786.52002)] who showed that the problem has a negative solution for sufficiently large \(n\). Next, the author recalls a number of estimates on the number of these parts. He also considers a related covering problem by \textit{H. Hadwiger} [``Ungelöste Probleme. 20'', Elem. Math. 12, 121 (1957)]. The last section of this article is entitled ``Computer program for Borsuk's conjecture''. It proposes to apply a net of lattice polytopes and the Hausdorff distances to any given set that is expected to be partitioned. The author adds that even for \(n=4\) ``it's to hard for todays computers'' since the unknown number of such lattice polytopes is at most \(e^{(\omega_4 \cdot 2760^4)^{3/5}}\), where \(\omega_4\) denotes the volume of the unit ball of \(E^4\).
reviewed by: Marek Lassak
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 52C45
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 7458828
zbMATH Keywords: Borsuk's problem
zbMATH Keywords: diameter
zbMATH Keywords: partition
zbMATH Keywords: lattice polytopes
Wikidata QID: Q113900229
author: Chuan Ming Zong
MaRDI profile type: Publication
full work available at URL: s11537-021-2007-7
OpenAlex ID: W3183651628
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DOI: 10.1007/S11537-021-2007-7
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description: scientific article; zbMATH DE number 7458828