Euclidean embeddings in spaces of finite volume ratio via random matrices

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 , || ⋅ ||) , i.e. there exist α , β > 0 such that for all x ∈ ℝ N , . This solves a conjecture of Schechtman on random embeddings of ℓ 2 n into ℓ 1 N .