Borsuk covering and planar sets with unique completion
type: Item
instance of: scholarly article
title: Borsuk covering and planar sets with unique completion
zbMATH Open document ID: 0792.52002
DOI: 10.1016/0012-365X(93)90298-8
author: Krzysztof Kołodziejczyk
published in: Discrete Mathematics
publication date: 1994-07-28T00:00:00Z
review text: A Borsuk covering of a bounded set \(S\) in \(\mathbb{R}^ n\) of diameter \(d(S)\) is a covering of \(S\) by a finite number of sets of diameter less than \(d(S)\). The Borsuk number of a set \(S\), denoted by \(b(S)\), is the spallest number of sets forming a Borsuk covering of \(S\). The number \(b(\mathbb{R}^ n)=\{\sup b(S):S\) bounded subset of \(\mathbb{R}^ n\}\) is called the Borsuk number of the space \(\mathbb{R}^ n\). The famous Borsuk problem is whether \(b(\mathbb{R}^ n)=n+1\). It is still open for \(n \geq 4\). In this paper an equivalent formulation of this problem is given. Namely, the following two assertions are equivalent: (i) \(b(\mathbb{R}^ n)=n+1\). (ii) For all convex bodies \(K\) and \(M\) of constant width in \(\mathbb{R}^ n\), we have \(b(K+M)=\{\min b(K),b(M)\}\).
Mathematics Subject Classification ID: 52A37
zbMATH DE Number: 500640
zbMATH Keywords: Borsuk problem
zbMATH Keywords: Borsuk numbers
zbMATH Keywords: constant width
reviewed by: J. Pečarić
MaRDI profile type: Publication
cites work: Convex functions and upper semi-continuous collections
cites work: O pазиени плоских фигур на части меньшего диаметра
cites work: Q3683105
cites work: Q3310510
cites work: Q3667511
cites work: Über Durchschnittseigenschaften n-dimensionaler Kugelfamilien.
cites work: Borsuk's covering for blunt bodies
cites work: Q3258669
cites work: Covering a Three-Dimensional set with Sets of Smaller Diameter
cites work: Q3237949
cites work: Q5530342
cites work: Q3283429
cites work: Q4007240
cites work: Q3660447
cites work: Zur Zerlegung von Punktmengen in solche kleineren Durchmessers
cites work: Q3278735
description: scientific article; zbMATH DE number 500640