New volume ratio properties for convex symmetric bodies in \({\mathbb{R}}^ n\)
type: Item
instance of: scholarly article
title: New volume ratio properties for convex symmetric bodies in \({\mathbb{R}}^ n\)
zbMATH Open document ID: 0617.52006
DOI: 10.1007/BF01388911
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143456
review text: Let K be any convex, centrally symmetric, bounded and absorbing set in \({\mathbb{R}}^ n\) and let \(K^*\) be the polar body of K. By a(K) denote the product of the volumes \(V(K)\cdot V(K^*)\). It is known that \(b^ 2_ n/n^{n/2}\leq a(K)\leq b^ 2_ n\), where \(b_ n\) denotes the volume of the (Euclidean) unit ball in \({\mathbb{R}}^ n\). The right hand side inequality is due to Blaschke and Santaló. It is attained precisely for ellipsoids. The left hand side inequality is due to Bambah. The authors improve this essentially by establishing the lower bound \(c^ nb^ 2_ n\), where c is an absolute positive constant (Theorem 1). This already comes very close to a conjecture of Mahler which asserts that \(4^ n/n!\) is the (optimal) lower bound. Given a normed space E with unit ball K, let v(E) denote the volume ratio \((V(K)/V(J))^{1/n}\), where j denotes John's ellipsoid of maximal volume contained in K. For cotype-2 spaces E with cotype-2 constant C(E) the following estimate for the volume ratio is established (Theorem 2): v(E)\(\leq cC(E) (\log C(E))^ 4\). This gives an affirmative answer to a question raised by Pelczynski. Theorem 1 admits interesting applications to the Geometry of Numbers: Let K be any bounded 0-symmetric convex body in \({\mathbb{R}}^ n\) and \(K^*\) its polar body. Then there exists an absolute constant \(c_ 1\) such that either \(c_ 1n^{1/2}K\) or \(c_ 1n^{1/2}K^*\) contains a non-trivial point of the integer lattice \({\mathbb{Z}}^ n\). This result is optimal for ellipsoids. Further consequences concern successive minima, mixed volumes and harmonic analysis.
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 43A25
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 4000872
zbMATH Keywords: polar bodies
zbMATH Keywords: Banach-Mazur distance
zbMATH Keywords: Rademacher projection
zbMATH Keywords: John ellipsoid
zbMATH Keywords: convex symmetric bodies
zbMATH Keywords: cotype-2 spaces
zbMATH Keywords: successive minima
zbMATH Keywords: mixed volumes
zbMATH Keywords: harmonic analysis
author: Jean Bourgain
author: Vitali Milman
reviewed by: Gerhard Ramharter
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4000872