Lower bounds for the numbers of antipodal pairs and strictly antipodal pairs of vertices in a convex polytope
type: Item
instance of: scholarly article
title: Lower bounds for the numbers of antipodal pairs and strictly antipodal pairs of vertices in a convex polytope
zbMATH Open document ID: 0799.52003
DOI: 10.1007/BF02574001
author: Valeriu Soltan
author: Nguyen Manh Hung
published in: Discrete \& Computational Geometry
publication date: 1994-11-09T00:00:00Z
full work available at URL: https://eudml.org/doc/131295
review text: Let \(X\) be a finite set of points in a \(d\)-dimensional space \(E^ d\). \textit{V. Klee} [`Unsolved problems in intuitive geometry,' Mimeographed notes, Seattle (1960)] calls points \(x\) and \(y\) from \(X\) antipodal if there are two distinct parallel hyperplanes \(H_ x\) and \(H_ y\) supporting the polytope \(P = \text{conv} X\) such that \(x\) and \(y\) belong to \(H_ x\) and \(H_ y\), respectively. If, additionally, \(P \cap H_ x = \{x\}\) and \(P \cap H_ y = \{y\}\) then the point pair \(x,y\) is called strictly antipodal [cf. \textit{B. Grünbaum}, Isr. J. Math. 1, 5-10 (1963; Zbl 0192.266)]. There are some results on upperbounds of points of a convex body or an \(n\)-pointed set, every pair of which is antipodal. In this paper the greatest lower bounds for the number of antipodal pairs \((r)\) (strictly antipodal pairs \((s))\) of points of a finite set in \(E^ d\) are determined. Avoiding degenerate situations it suffices to consider a convex polytope \(P_ n\) in \(E^ d\) with \(n\) vertices which describe that finite point set. For \(d=2\) the values \(r(P_ n)\) and \(s(P_ n)\) can be given exactly. From these an inequality of B. Grünbaum follows immediately. For \(P_ n\) in \(E^ d\), \(d \geq 3\) the authors give the inequalities \(r(P_ n) \geq n - 1 + d(d - 1)/2\) and \(s(P_{2k}) \geq k\), \(s(P_{2k + 1}) \geq k + 1\) and also all the cases for equalities. To prove this they consider families of convex cones with common zero apex and translate their problem of points into that of cones.
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 536872
zbMATH Keywords: convex polytopes
zbMATH Keywords: \(d\)-dimensional Euclidean space
zbMATH Keywords: lower bounds
zbMATH Keywords: antipodal points
reviewed by: Johannes Böhm
MaRDI profile type: Publication
cites work: Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V.L. Klee
cites work: Q3317690
cites work: Strictly antipodal sets
cites work: Q3974986
cites work: Irreducible convex sets
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description: scientific article; zbMATH DE number 536872