Characterizations of reduced polytopes in finite-dimensional normed spaces
type: Item
instance of: scholarly article
title: Characterizations of reduced polytopes in finite-dimensional normed spaces
zbMATH Open document ID: 1125.52010
author: Marek Lassak
published in: Beiträge zur Algebra und Geometrie
publication date: 2007-01-22T00:00:00Z
full work available at URL: https://eudml.org/doc/230908
full work available at URL: 18.html
review text: Let \(M^d\) be a \(d\)-dimensional normed real space. A strip \(S\) in \(M^d\) is the convex hull of two (distinct) parallel hyperplanes \(H_1\) and \(H_2\), its direction \(m\) is a (euclidean) perpendicular to the \(H_j\), and its width \(w(S)\) is the \(M^d\)-distance between \(H_1\) and \(H_2\). If \(H_1\) and \(H_2\) support a convex body \(C\), then \(S\) is a \(C\)-strip; if its direction is \(m\), then it is denoted by \(S(C,m)\), and \(w(S(C,m))\) is called the \(M^d\)-width of \(C\) in this direction. The number \(\Delta(C) = \min_mw(C,m)\) is the \(M^d\)-thickness of \(C\). If \(c_j \in H_j \cap C\) for \(j = 1,2\) and \(\| c_1 - c_2\| = \Delta(C)\), then \(c_1c_2\) is a thickness chord of \(C\). Denote by \(\Gamma(C,m)\) the \(M^d\)-length of the longest chord of \(C\) in direction \(m\), and set \(\Gamma(C) = \min_m\Gamma(C,m)\). Finally, a convex body \(R\) is called reduced if \(\Delta(K) < \Delta(R)\) whenever \(K \subset R\). The author proves here two equivalent conditions for a polytope \(P\) to be reduced. First, through each vertex \(v\) of \(P\) passes a strip of \(M^d\)-width \(\Delta(P)\) whose bounding hyperplane through \(v\) strictly supports \(P\) (that is, in \(v\) alone). Second, for each vertex \(V\) of \(P\) there is a direction such that the longest chord of \(P\) in this direction is unique, has \(M^d\)-length \(\Gamma(P)\) and \(v\) as its end-point. Also given are two characterizations of reduced simplices in \(M^d\), and one of reduced polygons in \(M^2\).
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 5117822
zbMATH Keywords: reduced polytope
zbMATH Keywords: normed space
zbMATH Keywords: width
zbMATH Keywords: thickness
zbMATH Keywords: chord
reviewed by: Peter McMullen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5117822