Two-point symmetrization and convexity
type: Item
instance of: scholarly article
title: Two-point symmetrization and convexity
zbMATH Open document ID: 1069.52012
author: G. Aubrun
author: Matthieu Fradelizi
published in: Archiv der Mathematik
publication date: 2004-11-26T00:00:00Z
review text: For a hyperplane \(H\) in \(\mathbb{R}^n\), let \(\sigma_H\) denote the reflection in \(H\), and let \(H^-\), \(H^+\) be the two closed halfspaces bounded by \(H\). The two-point-symmetrization with respect to \(H\) transforms a set \(A\subset \mathbb{R}^n\) into the set \[ ((A\cap \sigma_H A)\cap H^-)\cup((A\cup (\sigma_H A)\cap H^+). \] A similar definition (with great subspheres) is possible in spherical space \(\mathbb{S}^n\). The authors prove (in more general and refined versions) the following result. If \(K\) is an \(n\)-dimensional convex body (in \(\mathbb{R}^n\) or \(\mathbb{S}^n\)) such that every two-point symmetrization transforms \(K\) into a convex set, then \(K\) is a ball. This was conjectured by the reviewer and is used in a paper by \textit{F. Gao, D. Hug} and the reviewer [Math. Notae 41 (2001/02), 159--176 (2003)].
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52A55
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 2119265
zbMATH Keywords: two-point symmetrization
zbMATH Keywords: characterization of balls
MaRDI profile type: Publication
full work available at URL: s00013-003-4839-1
OpenAlex ID: W2104432796
DOI: 10.1007/S00013-003-4839-1
Recommended article: Symmetry and the farthest point mapping on convex surfaces
Recommended article: Symmetrization, convexity and applications
Recommended article: Symmetrization of convex plane curves
Recommended article: Symmetries of two-point sets
Recommended article: On the symmetry function of a convex set
Recommended article: On the structure of convex sets with symmetries
Recommended article: Convex symmetrization and applications
Recommended article: Points of symmetry of convex sets in the two-dimensional complex space -- a counterexample to D. Yost's problem
Recommended article: The critical point and related symmetry measures of a planar convex set
Recommended article: Q5208217
description: scientific article; zbMATH DE number 2119265