Note on bi-Lipschitz embeddings into normed spaces
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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.
[1] I. J. Schoenberg,et al. Metric spaces and positive definite functions , 1938 .
[2] J. Bourgain. On lipschitz embedding of finite metric spaces in Hilbert space , 1985 .
[3] Jean Bourgain,et al. On type of metric spaces , 1986 .
[4] J. Lindenstrauss,et al. On lipschitz embedding of finite metric spaces in low dimensional normed spaces , 1987 .
[5] J. Spencer. Ten lectures on the probabilistic method , 1987 .