Entropy extension
type: Item
instance of: scholarly article
title: Entropy extension
zbMATH Open document ID: 1118.52014
author: Alain Pajor
author: Nicole Tomczak-Jaegermann
author: Alexander E. Litvak
author: Vitali Milman
publication date: 2007-06-08T00:00:00Z
review text: One of the important consequences of the Hahn-Banach theorem is the so-called ``extension property of \(\ell_\infty\)''. It states that, given a normed space \(X\) and a subspace \(Y\subset X\), every linear operator \(S: Y\to\ell_\infty\) can be extended to an operator \(T: X\to\ell_\infty\) having the same norm as S. This theorem is used in the proofs of many results in Banach space theory and related fields. In particular, it was one of the ingredients of the following result on covering numbers, obtained recently in \textit{A.~E. Litvak, A. Pajor}, and \textit{N. Tomczak-Jaegermann} [J. Funct. Anal. 231, No.~2, 438--457 (2006; Zbl 1092.52004)]: Let \(0 < a < r < A\) and \(1\leqslant k < n\). Let \(K, L\subset\mathbb R^n\) be symmetric convex bodies, and let \(K\subset AL\). Let \(E\subset\mathbb R^n\) be a \(k\)-codimensional subspace such that \(K\cap E\subset aL\). Then \[ N(K, 2rL)\leqslant 2^k\left(\frac{A+r}{r-a}\right)^k, \] where \(N(K,L)\) denotes the covering number. In some sense, the last result is a (weak) version of the extension theorem for entropy: if we control the norm of the identity operator (\(=\) half diameter of the unit ball) in a subspace, then we control entropy in the entire space. Note that if \(K\cap E\subset aL\), then trivially \(N(K\cap E, aL)\leqslant1\). However, why should the diameter play such a crucial role? Can one achieve a similar control of entropy in the entire space from the knowledge of entropy (rather than the diameter) in a subspace? Intuition does not support such a hope. However, quite surprisingly, this is possible. The authors prove a strong version of an extension theorem for entropy: if we control entropy in a subspace, then we control entropy in the entire space. The main result of this paper is the ``entropy extension-lifting theorem.'' It consists of two inequalities for entropies. The first inequality relates the entropy of \(K\) and \(L\) to the entropy of sections of small codimension and can be called an…
reviewed by: Serguey M. Pokas
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 54C70
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 46B07
zbMATH DE Number: 5163307
zbMATH Keywords: metric entropy
zbMATH Keywords: Hahn-Banach theorem
zbMATH Keywords: entropy extension-lifting theorem
zbMATH Keywords: entropy decomposition
zbMATH Keywords: covering numbers
zbMATH Keywords: nonsymmetric body
MaRDI profile type: Publication
full work available at URL: s10688-006-0046-8
OpenAlex ID: W3190096422
cites work: Diameters of sections and coverings of convex bodies
cites work: On the Euclidean Metric Entropy of Convex Bodies
cites work: A note on coverings
cites work: Q5568974
cites work: Covering convex bodies by translates of convex bodies
DOI: 10.1007/S10688-006-0046-8
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 5163307