Israel Journal of Mathematics

, Volume 93, Issue 1, pp 333–344

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

Article

DOI: 10.1007/BF02761110

Cite this article as:
Matoušek, J. Israel J. Math. (1996) 93: 333. doi:10.1007/BF02761110

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)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 thelk 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.

Copyright information

© The magnes press 1996

Authors and Affiliations

  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic

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