Random Euclidean embeddings in spaces of bounded volume ratio

Abstract

Let ( R N,‖·‖) be the space R 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 ( R n,|·|) onto its image in ( R N,‖·‖) : there exist α, β>0 such that for all x∈ R n , α N |x|⩽‖Γx‖⩽β N |x| . This solves a conjecture of Schechtman on random embeddings of ℓ 2 n into ℓ 1 N . To cite this article: A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).