On the asymmetry constant of a body with few vertices
type: Item
instance of: scholarly article
title: On the asymmetry constant of a body with few vertices
zbMATH Open document ID: 0999.52008
DOI: 10.1023/A:1014994419758
published in: Geometriae Dedicata
publication date: 2002-05-30T00:00:00Z
review text: Introducing the distance measure \[ d(K,L)=\text{inf}\{\alpha\beta: \alpha>0,\;\beta>0, \frac 1\beta L\subset K\subset \alpha L\} \] for convex bodies \(K,L\subset\mathbb{R}^n\), the authors consider the asymmetry constant \[ \sigma(K) = d(K,C^n) = \text{inf}\{d(K-a,B): a\in\mathbb{R}^n,\;B\subset C^n\} \] of a convex body \(K\subset \mathbb{R}^n\), where \(C^n\) denotes the family of all convex bodies centred at the origin. They are concerned with the problem of how small could be the asymmetry constant of convex \(n\)-dimensional polytopes between (regarding the cardinality of the vertex set) the studied extremal case of a simplex and that of a cross polytope. It is shown that any convex polytope \(P\subset \mathbb{R}^n\) with \(n+k\) vertices, \(1\leq k<n\), is far from any symmetric polytope, and asymptotically sharp estimates for the asymmetry constant of such polytopes are provided.
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 46B07
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1747819
zbMATH Keywords: measure of symmetry
zbMATH Keywords: measure of asymmetry
zbMATH Keywords: simplex
zbMATH Keywords: cross polytope
zbMATH Keywords: convex polytope
author: Efim D. Gluskin
author: Alexander E. Litvak
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1747819