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 k f k Lip = sup{ρ(f(x), f(y))/d(x, y); x, y ∈ X, x 6 y}, and dist(f) = k f k Lip.k f 1 k 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 l N with a prescribed distortion D. We obtain that this is possible for N ≥ C(log n)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 lN are obtained by a similar method.