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finite metric space; embedding of metric spaces; distortion; Lipschitz mapping; spaces $\ell_p$
Let $(X,d)$, $(Y,\rho)$ be metric spaces and $f:X\to Y$ an injective mapping. We put $\|f\|_{\operatorname{Lip}} = \sup \{\rho (f(x),f(y))/d(x,y); x,y\in X, x\neq y\}$, and $\operatorname{dist}(f)= \|f\|_{\operatorname{Lip}}.\| f^{-1}\|_{\operatorname{Lip}}$ (the {\sl distortion} of the mapping $f$). We investigate the minimum dimension $N$ such that every $n$-point metric space can be embedded into the space $\ell_{\infty }^N$ with a prescribed distortion $D$. We obtain that this is possible for $N\geq C(\log n)^2 n^{3/D}$, 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 $\ell_p^N$ are obtained by a similar method.
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[JLS87] Johnson W., Lindenstrauss J., Schechtman G.: On Lipschitz embedding of finite metric spaces in low dimensional normed spaces. in: {\sl Geometrical aspects of functional analysis} (J. Lindenstrauss, V.D. Milman eds.), Lecture Notes in Mathematics 1267, Springer-Verlag, 1987. MR 0907694 | Zbl 0631.46016
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[Scho38] Schoenberg I.J.: Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44 (1938), 522-536. MR 1501980 | Zbl 0019.41502
[Spe87] Spencer J.: Ten Lectures on the Probabilistic Method. CBMS-NSF, SIAM 1987. MR 0929258 | Zbl 0822.05060
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