Random ε-nets and embeddings in lN∞

Abstract

We show that, given an n-dimensional normed space X, a sequence of N = (8/e) 2n independent random vectors (Xi) N=1, uniformly distributed in the unit ball of X � , with high probability forms an e-net for this unit ball. Thus the random linear map Γ : R n → R N defined by Γ x = (h x, Xii ) N=1 embeds X in l N with at most 1 + e norm distortion. In the case X = l n we obtain a random 1 + e-embedding into l N with asymptotically best possible relation between N, n, and e. 1. Introduction. Let X = (R n , kk ) be an arbitrary n-dimensional normed space with unit ball K. It is well known that, for any 0 < e < 1, X can be 1 + e-embedded into l N, for some N = N(e, n), depending on e and n, but independent of X. In this note we investigate 1+e-isomorphic embed- dings which are random with respect to some natural measure, depending on K. We first show that for N = (8/e) 2n , a sequence of N independent ran- dom vectors (Xi) N=1 , uniformly distributed in the unit ball K 0 of the dual space X ∗ , forms an e-net for K 0 with high probability. Thus, with high prob- ability, the random linear map Γ : R n → R N defined by Γ x = (h x, Xii ) N=1 embeds X in l N with at most 1 + e norm distortion. The important case is X = l n . In this case it is more natural to consider random vectors Xi uniformly distributed on the sphere S n−1 . Such vectors also form an e-net on the sphere, hence they determine a random 1 + e- embedding Γ of l n into l N. We also show that p n/N Γ is a 1+ e-isometry from l n into l N , with high probability. The case X = l n is connected with Dvoretzky's theorem ((D)). Milman found a new proof ((M)), using the Levy isoperimetric inequality on the