Let be the space 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 onto its image in : there exist α,β>0 such that for all , . This solves a conjecture of Schechtman on random embeddings of ℓ2n into ℓ1N.
Soit l'espace 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 : il existe α,β>0 tels que pour tout , ; ce qui démontre une conjecture de Schechtman sur les plongements aléatoires de ℓ2n dans ℓ1N.
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Alexander Litvak 1; Alain Pajor 2; Mark Rudelson 3; Nicole Tomczak-Jaegermann 1; Roman Vershynin 4
@article{CRMATH_2004__339_1_33_0, author = {Alexander Litvak and Alain Pajor and Mark Rudelson and Nicole Tomczak-Jaegermann and Roman Vershynin}, title = {Random {Euclidean} embeddings in spaces of bounded volume ratio}, journal = {Comptes Rendus. Math\'ematique}, pages = {33--38}, publisher = {Elsevier}, volume = {339}, number = {1}, year = {2004}, doi = {10.1016/j.crma.2004.04.019}, language = {en}, }
TY - JOUR AU - Alexander Litvak AU - Alain Pajor AU - Mark Rudelson AU - Nicole Tomczak-Jaegermann AU - Roman Vershynin TI - Random Euclidean embeddings in spaces of bounded volume ratio JO - Comptes Rendus. Mathématique PY - 2004 SP - 33 EP - 38 VL - 339 IS - 1 PB - Elsevier DO - 10.1016/j.crma.2004.04.019 LA - en ID - CRMATH_2004__339_1_33_0 ER -
%0 Journal Article %A Alexander Litvak %A Alain Pajor %A Mark Rudelson %A Nicole Tomczak-Jaegermann %A Roman Vershynin %T Random Euclidean embeddings in spaces of bounded volume ratio %J Comptes Rendus. Mathématique %D 2004 %P 33-38 %V 339 %N 1 %I Elsevier %R 10.1016/j.crma.2004.04.019 %G en %F CRMATH_2004__339_1_33_0
Alexander Litvak; Alain Pajor; Mark Rudelson; Nicole Tomczak-Jaegermann; Roman Vershynin. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.04.019/
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