Functional Analysis/Probability Theory
Random Euclidean embeddings in spaces of bounded volume ratio
Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 33-38.

Let ( N ,·) be the space N equipped with a norm ‖·‖ whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N×n matrix with N>n, whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n-dimensional Euclidean space ( n ,|·|) onto its image in ( N ,·): there exist α,β>0 such that for all x n , αN|x|ΓxβN|x|. This solves a conjecture of Schechtman on random embeddings of ℓ2n into ℓ1N.

Soit ( N ,·) l'espace N muni d'une norme ‖·‖ dont la boule unité est à volume ratio borné par rapport à la boule unité euclidienne. On montre qu'une matrice aléatoire Γ, de taille N×n (N>n), dont les coefficients sont des variables aléatoires indépendantes, vérifiant certaines hypothèses de moments, réalise avec une grande probabilité, un bon isomorphisme de l'espace euclidien de dimension n, de norme |·|, sur son image dans ( N ,·) : il existe α,β>0 tels que pour tout x n , αN|x|ΓxβN|x| ; ce qui démontre une conjecture de Schechtman sur les plongements aléatoires de ℓ2n dans ℓ1N.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.04.019
Litvak, Alexander 1; Pajor, Alain 2; Rudelson, Mark 3; Tomczak-Jaegermann, Nicole 1; Vershynin, Roman 4

1 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
2 Équipe d'analyse et mathématiques appliquées, université de Marne-la-Vallée, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée cedex 2, France
3 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
4 Department of Mathematics, University of California, Davis, CA 95616, USA
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Litvak, Alexander; Pajor, Alain; Rudelson, Mark; Tomczak-Jaegermann, Nicole; Vershynin, Roman. Random Euclidean embeddings in spaces of bounded volume ratio. Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 33-38. doi : 10.1016/j.crma.2004.04.019. http://www.numdam.org/articles/10.1016/j.crma.2004.04.019/

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