Stability theorems for two measures of symmetry
type: Item
instance of: scholarly article
title: Stability theorems for two measures of symmetry
zbMATH Open document ID: 0960.52005
published in: Discrete \& Computational Geometry
publication date: 2001-05-10T00:00:00Z
review text: The paper proposes two new measures of symmetry of a convex body \(K\) in Euclidean \(n\)-space \(\mathbb{R}^n\). The first measure is \(\eta(K)= \sup\{h_K (u)/h_K (-u):u\in S^{n-1}\}\), where \(h_K(u)\) denotes the support function of \(K\) with respect to the centroid of \(K\). The second measure is \(\theta(K)=\text{sup\{vol}(K_u)/\text{vol}(K_{-u}):u\in S^{n-1}\}\) with \(K_u=K\cap E^+(u)\), where \(E^+(u)\) denotes the half-space whose boundary hyperplane contains the centroid of \(K\) and has \(u\) as an interior normal vector. Cones appear to be the most asymmetric convex bodies for both the measures. Here a cone means the convex hull of the union of a convex compact subset of a hyperplane and of a point out of this hyperplane. The author establishes stability estimates for both measures which say about the deviation of a convex body from a cone. For instance, the estimate for the measure \(\eta\) says that for every \(\varepsilon\geq 0\) such that \(\eta(K)\geq n-\varepsilon\) there exists a cone \(C\) such that \(\text{vol} (K\cup C)-\text{vol}(K\cap C)\leq {4\over n+1}\text{vol} (K)\varepsilon\).
reviewed by: Marek Lassak
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 1507677
zbMATH Keywords: measure of symmetry
zbMATH Keywords: cone
zbMATH Keywords: stability estimates
zbMATH Keywords: deviation
author: Helmut Groemer
MaRDI profile type: Publication
full work available at URL: s004540010037
OpenAlex ID: W1984110978
DOI: 10.1007/S004540010037
description: scientific article; zbMATH DE number 1507677