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

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Abstract

We 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)n 1/2D for somec(D)>0. For aD-embedding of alln-point metric spaces into the samek-dimensional normed spaceX we find an upper boundk≤12Dn 1/[(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.

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Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří Matoušek

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  1. Jiří Matoušek
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Correspondence to Jiří Matoušek.

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The research was supported by Charles University grant No. 351, by Czech Republic Grant GAČR 201/93/2167 and by EC Cooperative Action IC-1000 (project ALTEC:Algorithms for Future Technologie).

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Matoušek, J. On the distortion required for embedding finite metric spaces into normed spaces. Israel J. Math. 93, 333–344 (1996). https://doi.org/10.1007/BF02761110

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  • DOI: https://doi.org/10.1007/BF02761110

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