Borsuk's covering for blunt bodies
type: Item
instance of: scholarly article
title: Borsuk's covering for blunt bodies
zbMATH Open document ID: 0628.52015
DOI: 10.1007/BF01194159
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Denote by \(d_ n\) the smallest number such that there exists a covering of the unit sphere \(S^{n-1}\) by \(n+1\) sets each of diameter \(\leq d_ n\). (Thus \(d_ 2=2\pi /3).)\) Let C be a convex body in \(E^ n\) and let \(x\in \partial C\). Denote by \(S(x,C)\subset S^{n-1}\) the set of directions of all diameters of C starting from x. Suppose that the circumradius of \(S(x,C)\) in \(S^{n-1}\) is \(<(\pi -d_ n)\) for any \(x\in \partial C\) where \(S(x,C)\neq \emptyset\). Then C can be covered by \(n+1\) sets, each of diameter less than diameter \(C.\)
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 52A55
zbMATH DE Number: 4022043
zbMATH Keywords: Borsuk's problem
zbMATH Keywords: convex bodies
author: B. V. Dekster
MaRDI profile type: Publication
cites work: Convex functions and upper semi-continuous collections
cites work: Covering a Three-Dimensional set with Sets of Smaller Diameter
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cites work: Überdeckung einer Menge durch Mengen kleineren Durchmessers
cites work: Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers
cites work: Über die Zerstückung eines Eikörpers
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cites work: Konstruktion regulärer Hüllen konstanter Breite
cites work: Eine Bemerkung zur Überdeckung beschränkter Mengen durch Mengen kleineren Durchmessers
full work available at URL: https://doi.org/10.1007/bf01194159
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description: scientific article; zbMATH DE number 4022043