Measures of symmetry for convex sets and stability
type: Item
instance of: scholarly article
title: Measures of symmetry for convex sets and stability
zbMATH Open document ID: 1335.52002
DOI: 10.1007/978-3-319-23733-6
author: Gábor Tóth
published in: Universitext
publication date: 2015-10-08T00:00:00Z
review text: This is a graduate-level textbook on convex geometry in finite-dimensional Euclidean spaces, which has some interesting special features. Using the author's words, it starts from the beginning, guides the reader through the necessary background, and ends up in recent research. A special feature is that the main part of the book is developed along two particular, though comprehensive, topics of classical convex geometry, namely measures of symmetry, and stability of geometric inequalities. A measure of symmetry is a continuous real function on the space of convex bodies, perhaps with specific additional properties, that attains one of its extrema precisely at symmetric bodies, so that this function generally measures the deviation from symmetric shape. A stability result for a geometric inequality for convex bodies estimates how much a convex body deviates from an extremal body if it satisfies the inequality with approximate equality. The well-known survey articles by \textit{Branko Grünbaum} on ``Measures of symmetry for convex sets'' [Proc. Sympos. Pure Math. 7, 233--270 (1963; Zbl 0142.20503)] and by \textit{Helmut Groemer} on ``Stability of geometric inequalities'' [in: Handbook of convex geometry. Volume A. Amsterdam: North-Holland. 125--150 (1993; Zbl 0789.52001)] showed impressively how rich these topics were already decades ago. The first chapter of this book lays the necessary foundations on convex sets in finite dimensions and treats, for example, the Hausdorff distance (and other distances), the Blaschke selection theorem, separation theorems, Minkowski's theorem on extreme points, mixed volumes (briefly), the combinatorial theorems of Carathéodory, Radon and Helly, and classical results on circumradius and inradius. The chapter concludes with a detailed proof, using Painlevé-Kuratowski convergence, of a Helly-type theorem of Klee on unbounded convex sets. The second chapter treats various notions of immediate geometric appeal, but also preparing the study…
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52-02
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A37
Mathematics Subject Classification ID: 52-01
zbMATH DE Number: 6492282
zbMATH Keywords: convex geometry
zbMATH Keywords: measure of symmetry
zbMATH Keywords: stability of geometric inequalities
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-23733-6
OpenAlex ID: W2290236472
description: scientific article; zbMATH DE number 6492282