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Israel Journal of Mathematics

, Volume 93, Issue 1, pp 333–344 | Cite as

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

  • Jiří MatoušekEmail author
Article

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.

Keywords

Normed Space Complete Bipartite Graph Real Algebraic Variety Finite Projective Plane Ramanujan Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© The magnes press 1996

Authors and Affiliations

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

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