High dimensional random sections of isotropic convex bodies
type: Item
instance of: scholarly article
title: High dimensional random sections of isotropic convex bodies
zbMATH Open document ID: 1221.52007
author: David Alonso-Gutiérrez
author: Jesús Bastero
author: Julio Bernués
author: Grigoris Paouris
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-11-10T00:00:00Z
review text: The authors study two properties of random high dimensional convex bodies. In the first part they consider a symmetric convex body \(K\) and they estimate the central section function \(|K\cap F^{\bot}|_{n-k}^{1/k}\), for random \(F\subset G_{n,k}\), \(1\leq k\leq n,\) where \(G_{n,k}\) is the Grassmann space. In [Lect. Notes Math. 1376, 64--104 (1989; Zbl 0679.46012), p. 88], \textit{V. D. Milman} and \textit{A. Pajor} show that \(|K\cap F^{\bot}|_{n-k}^{1/k}\sim L_{B_{k+1}}(K,F)/L_K\), where \(L_K\) is the isotropy constant of \(K\), \(B_{k+1}(K,F)\) the unit ball with norm \(\phi\rightarrow|\phi|^{k+2}_{(K+2)}\left(\int_{K\cap E(\phi)}|\chi,\phi|^{k+1}d\chi\right)^{\frac{-1}{(k+2)}}.\) This result holding for each subspace \(F\in G_{n,k}\) is generalized by the authors for each \textit{most} subspace. In the second part for every symmetric convex body, a bound for volume ratio of random high dimensional sections \(F\in G_{n,k}\) is given.
reviewed by: Loredana Sorrenti
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 5629198
zbMATH Keywords: isotropic convex bodies
zbMATH Keywords: section function
zbMATH Keywords: \(M\)-position
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.07.011
OpenAlex ID: W2065216966
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DOI: 10.1016/J.JMAA.2009.07.011
description: scientific article; zbMATH DE number 5629198