The Banach-Mazur distance to the cube in low dimensions
type: Item
instance of: scholarly article
title: The Banach-Mazur distance to the cube in low dimensions
zbMATH Open document ID: 1223.52001
author: Steven Taschuk
published in: Discrete \& Computational Geometry
publication date: 2011-06-10T00:00:00Z
review text: The Banach-Mazur distance between two symmetric convex bodies \(K,L\subset \mathbb{R}^n\) is defined by \(d(K,L):= \inf\{\lambda>0: K\subseteq TL\subseteq \lambda K \; \text{for} \; T\in GL(\mathbb{R}^n)\}\). \textit{M. Lassak} [Bull. Pol. Acad. Sci., Math. 39, No. 3--4, 219--223 (1991; Zbl 0756.52007)] proved the upper bound \(d(K,B^n) \leq \sqrt{n^2-n+1}\) for symmetric convex bodies \(K\) (\(K=-K\)) and the \(n\)-dimensional cube \(B^n=[-1,1]^n\). The main result of the present paper is: \(d(K,B^n)\leq \sqrt{n^2-2n+2 +2/(\sqrt{n+2}-1)}\) for general \(n\); this improves Lassak's upper bound for \(n\geq 3\), but for large \(n\), asymptotically better bounds are known.
reviewed by: Eike Hertel
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A27
Mathematics Subject Classification ID: 52A21
zbMATH DE Number: 5906691
zbMATH Keywords: symmetric convex bodies
zbMATH Keywords: Banach-Mazur distance
MaRDI profile type: Publication
full work available at URL: s00454-010-9251-6
OpenAlex ID: W2054944142
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DOI: 10.1007/S00454-010-9251-6
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description: scientific article; zbMATH DE number 5906691