Note on bi-Lipschitz embeddings into normed spaces

Let (X, d), (Y, ρ) be metric spaces and f : X → Y an injective mapping. We put ‖f‖Lip = sup{ρ(f(x), f(y))/d(x, y); x, y ∈ X, x 6= y}, and dist(f) = ‖f‖Lip.‖f ‖Lip (the distortion of the mapping f). We investigate the minimum dimension N such that every n-point metric space can be embedded into the space lN ∞ with a prescribed distortion D. We obtain that this is possible for N ≥ C(logn)2n3/D, where C is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into lp are obtained by a similar method.