论文信息 - On the distortion required for embedding finite metric spaces into normed spaces

On the distortion required for embedding finite metric spaces into normed spaces

AbstractWe investigate the minimum dimensionk such that anyn-point metric spaceM can beD-embedded into somek-dimensional normed spaceX (possibly depending onM), that is, there exists a mappingf: M→X with $$\frac{1}{D}dist_M (x,y) \leqslant \left| {f(x) - f(y)} \right| \leqslant dist_M (x,y) for any$$ Extending a technique of Arias-de-Reyna and Rodríguez-Piazza, we prove that, for any fixedD≥1,k≥c(D)n1/2D for somec(D)>0. For aD-embedding of alln-point metric spaces into the samek-dimensional normed spaceX we find an upper boundk≤12Dn1/[(D+1)/2]lnn (using thel∞k space forX), and a lower bound showing that the exponent ofn cannot be decreased at least forD∃[1,7)∪[9,11), thus the exponent is in fact a jumping function of the (continuously varied) parameterD.

J. Matousek | J. Matoušek

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