Fine structure of convex sets from asymmetric viewpoint
type: Item
instance of: scholarly article
title: Fine structure of convex sets from asymmetric viewpoint
zbMATH Open document ID: 1225.52004
author: Gábor Tóth
published in: Beiträge zur Algebra und Geometrie
publication date: 2011-05-16T00:00:00Z
review text: Given a convex body \({K}\) in the Euclidean space, the author defines a sequence of measures of the symmetry of \(K\) with respect to a fixed interior point \(O \in {\text{Int}}(K)\). Although it is not the original definition the author starts with, all terms of the above sequence are subsequently shown to be obtained from an affine invariant function \(\sigma\) defined on certain slices of \(K\). More precisely, \(\sigma_m(K)=\inf_{O\in F,\;{\text{dim}} F=m} \sigma (K \cap F)\), where \(F\) denotes an \(m\)-dimensional affine space passing through \(O\). The paper contains several results relating the sequence \(\sigma_m\) to the geometric properties of the body \(K\). In particular, the symmetry of \(K\) with respect to \(O\) implies that the singular set of the function \(\sigma\) consists of all the interior points of \(K\). It is shown, via constructive examples, that the converse does not hold. In general, to calculate the singular set of \(\sigma\) is difficult, hence the author provides a number of results which facilitate this task.
reviewed by: Alina Stancu
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 5888457
zbMATH Keywords: convex bodies
zbMATH Keywords: convex polytopes
zbMATH Keywords: distortion
zbMATH Keywords: measure of symmetry
MaRDI profile type: Publication
full work available at URL: s13366-011-0019-9
OpenAlex ID: W1985090866
cites work: Carathéodory's theorem and \(H\)-convexity
cites work: Q5517554
cites work: Q4434671
cites work: On a problem of Steinhaus
cites work: Simplicial intersections of a convex set and moduli for spherical minimal immersions
cites work: On the shape of the moduli of spherical minimal immersions
cites work: On the structure of convex sets with applications to the moduli of spherical minimal immersions
cites work: On the structure of convex sets with symmetries
cites work: Asymmetry of convex sets with isolated extreme points
DOI: 10.1007/S13366-011-0019-9
description: scientific article; zbMATH DE number 5888457