Duality of metric entropy in Euclidean space.
type: Item
instance of: scholarly article
title: Duality of metric entropy in Euclidean space.
zbMATH Open document ID: 1042.52002
author: Stanislaw J. Szarek
author: S. Artstein-Avidan
author: Vitali Milman
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-01-28T00:00:00Z
review text: The subject of this paper is the duality conjecture for entropy numbers going back to \textit{A. Pietsch} [Theorie der Operatorenideale, Friedrich-Schiller-Universität Jena (1972; Zbl 0238.46067)]. For two convex bodies \(K\) and \(T\) in \({\mathbb R}^{n}\), the covering number of \(K\) by \(T\), denoted by \(N(K,T)\), is defined as the minimal number of translates of \(T\) needed to cover \(K\). The conjecture says that there exist two universal constants \(a,b\geq 1\) such that for any dimension \(n\) and for any two \(o\)-symmetric convex bodies \(K\) and \(T\) in \({\mathbb R}^{n}\) one has \(\log N(T^{\circ },aK^{\circ })\leq b\log N(K,T)\) where \(A^{\circ }\) denotes the polar body of \(A\). Here this conjecture is asserted to be valid in the important case where one of the two bodies is an ellipsoid. There is a sketch of a proof, but the details are announced to appear elsewhere.
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B07
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 2034210
zbMATH Keywords: covering number
zbMATH Keywords: entropy numbers
zbMATH Keywords: duality conjecture
zbMATH Keywords: ellipsoid
reviewed by: Johann Linhart
MaRDI profile type: Publication
full work available at URL: j.crma.2003.09.033
OpenAlex ID: W2044836164
cites work: Proportional concentration phenomena on the sphere
cites work: More on the duality conjecture for entropy numbers.
cites work: Q3761216
cites work: Q3124805
cites work: Q2702149
cites work: Q5648969
cites work: Q3761140
DOI: 10.1016/J.CRMA.2003.09.033
description: scientific article; zbMATH DE number 2034210