On the structure of convex sets with symmetries
type: Item
instance of: scholarly article
title: On the structure of convex sets with symmetries
zbMATH Open document ID: 1189.52007
author: Gábor Tóth
published in: Geometriae Dedicata
publication date: 2010-01-29T00:00:00Z
review text: The author studies a measure of symmetry \(\sigma\) associated to an \(n\)-dimensional convex body \(\mathcal L\) and a point \(\mathcal O\) in the interior of \(\mathcal L\). This measure is introduced as follows. If \(C\) is a boundary point of \(\mathcal L\), let \(C^0\) be the boundary point of \(\mathcal L\) distinct from \(\mathcal O\) and such that \(C, \mathcal O\) and \(C^0\) are collinear. By \(\Lambda(C,\mathcal O)\) we denote the ratio of the distance between \(C\) and \(\mathcal O\) to the distance between \(C^0\) and \(\mathcal O\). Now consider the convex hull \(P\) of points \(C_0,\ldots,C_n\) on the boundary of \(\mathcal L\) such that \(\mathcal O \in P\). Then \(\sigma(\mathcal L,\mathcal O)\) is defined as the infimum of (\({c_{i} val}\)) \(\sum_{i=0}^n \frac{1}{1+\Lambda(C_{i},\mathcal O)}\) over all \(C_0,\ldots,C_n\) as above. It can be shown that \(1 \leq \sigma(\mathcal L,\mathcal O) \leq \frac{n+1}{2}\) with the equality on the left attained if and only if \(\mathcal L\) is a simplex and inequality on the right attained if and only if \(\mathcal L\) is centrally symmetric with respect to \(\mathcal O\). The definition of \(\sigma\) allows to split the interior of the convex body \(\mathcal L\) into two classes of points. The point \(\mathcal O\) is said to be a \textit{regular point} of \(\mathcal L\) if equality of (ref \({c_i val}\)) and \(\sigma(\mathcal L,\mathcal O)\) implies that \(P\) (defined above) is a simplex and \(\mathcal O\) is in the interior of \(P\). Otherwise, \(\mathcal O\) is called a \textit{singular point} of \(\mathcal L\). The author shows that simplices (i.e., bodies \(\mathcal L\) with \(\sigma(\mathcal L,\mathcal O)=1\)) have interior consisting of regular points only and centrally symmetric convex bodies (i.e., bodies \(\mathcal L\) with \(\sigma(\mathcal L,\mathcal O)=(n+1)/2\)) have interior consisting of singular points only. (This result follows from a more general theorem by the author.) The author conjectur…
reviewed by: Gennadiy Averkov
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 5664573
zbMATH Keywords: convex set
zbMATH Keywords: measure of symmetry
MaRDI profile type: Publication
full work available at URL: s10711-009-9373-2
OpenAlex ID: W2074271882
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DOI: 10.1007/S10711-009-9373-2
description: scientific article; zbMATH DE number 5664573