Geometry of spaces between polytopes and related zonotopes
type: Item
instance of: scholarly article
title: Geometry of spaces between polytopes and related zonotopes
zbMATH Open document ID: 1032.46016
DOI: 10.1016/S0007-4497(02)01139-9
author: Elisabeth M. Werner
author: Yehoram Gordon
author: Alexander E. Litvak
author: Carsten Schütt
published in: Bulletin des Sciences Mathématiques
publication date: 2003-03-26T00:00:00Z
review text: Let \(k\leq N\) be positive integers, \((a_i)_{i\leq\mathbb{N}}\subset \mathbb{R}^n\) be a sequence of vectors spanning \(\mathbb{R}^n\), and let \(\|x\|_{k,q}= (\sum_{1\leq i\leq k}|\langle x,a_i\rangle|^{*q})^{{1\over q}}\), where \((|\langle x,a_i\rangle|^*)_i\) is the nonincreasing rearrangement of the sequence \((|\langle x,a_i\rangle|)_i\). The normed space \((\mathbb{R}^n,\|\cdot\|_{k,q})\) is denoted by \(X_{k,q}\), its unit ball is denoted by \(B_{k,q}\). The authors investigate the geometry of these spaces and their duals. These spaces are of interest because they generalize the class of dual spaces of zonotopes in a natural way. For \(k= N\) and \(q= 1\), the unit balls of \(X^*_{k,q}\) are zonotopes, and for \(k= 1\) and \(q= 1\), the spaces \(X_{k,q}\) range over all possible spaces with a polytopal unit ball with no more than \(2N\) facets. The authors provide upper and lower estimates for the volumes of \(B_{k,q}\) and their lower-dimensional sections. They determine the dimension of almost Euclidean sections obtaining a Dvoretzky-type theorem. Dealing with random parameters of the spaces, the authors apply the probabilistic results obtained by them in [Ann. Probab. 30, 1833-1853 (2002; Zbl 1016.60008)]. In the last section, the authors consider the important particular case when \(N= n\), \((a_i)_i= (e_i)_i\) is the canonical basis of \(\mathbb{R}^n\), and \(q= 1\). In this case, the space \(X_{k,1}\) becomes an interpolation space between \(\ell_1\) and \(\ell_\infty\). The authors obtain asymptotically sharp estimates for the most important geometric parameters of \(X_{k,1}\) such as type and cotype constants, projection constant, volume ratio, \(p\)-summing norms, etc.
Mathematics Subject Classification ID: 46B07
Mathematics Subject Classification ID: 46B09
Mathematics Subject Classification ID: 46B45
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A21
zbMATH DE Number: 1888298
zbMATH Keywords: Dvoretzky's theorem
zbMATH Keywords: \(p\)-summing norm
zbMATH Keywords: Random variables
zbMATH Keywords: Volume ratio
zbMATH Keywords: Polytopes
zbMATH Keywords: Zonoids
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1888298