Geometric and probabilistic analysis of convex bodies with unconditional structures, and associated spaces of operators
type: Item
instance of: scholarly article
title: Geometric and probabilistic analysis of convex bodies with unconditional structures, and associated spaces of operators
zbMATH Open document ID: 1216.46010
author: Yehoram Gordon
author: Mathieu Meyer
published in: Positivity
publication date: 2010-04-13T00:00:00Z
review text: The paper deals with geometric properties of unconditional and symmetric norms on \(\mathbb{R}^n\). It is proved that if \(\|\cdot\|\) is unconditional (with respect to the canonical basis \((e_i)_i\)) and \(\|e_i\|=1\) for every \(i\) then for every \(x\) one has \(2\|x\|_2 \leq \|x\| + \|x\|_* \leq \|x\|_1 + \|x\|_{\infty}\), where \(\|x\|_p\) denotes the \(\ell _p\)-norm. If \(\|\cdot\|\) is in addition symmetric then for every \(x\), \(y\) one has \(\|x\| \|y\|_* \leq \max\{\|x\|_1 \|y\|_{\infty},\;\|x\|_{\infty} \|y\|_1\}\). The authors provide many examples and some extensions to the case of two symmetric norms \(\|\cdot\|\) and \(|||\cdot|||\) satisfying \(\|\cdot\|\leq |||\cdot|||\leq \|\cdot\|_*\). They also show applications to the random embeddings of Euclidean spaces into spaces of nuclear operators and to the behavior of the so-called \(\ell\)-norms of operators.
Mathematics Subject Classification ID: 46B06
Mathematics Subject Classification ID: 46B07
Mathematics Subject Classification ID: 46B09
Mathematics Subject Classification ID: 46B28
zbMATH DE Number: 5692402
zbMATH Keywords: convex bodies
zbMATH Keywords: \(\ell\)-norm
zbMATH Keywords: nuclear operators
zbMATH Keywords: random Gaussian operator
zbMATH Keywords: symmetric norm
zbMATH Keywords: unconditional norm
MaRDI profile type: Publication
full work available at URL: s11117-009-0007-6
OpenAlex ID: W2047855300
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DOI: 10.1007/S11117-009-0007-6
description: scientific article; zbMATH DE number 5692402