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On dominatedl 1 metrics

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Abstract

We introduce and study a classl dom1 (ρ) ofl 1-embeddable metrics corresponding to a given metric ρ. This class is defined as the set of all convex combinations of ρ-dominated line metrics. Such metrics were implicitly used before in several constuctions of low-distortion embeddings intol p -spaces, such as Bourgain’s embedding of an arbitrary metric ρ onn points withO(logh) distortion. Our main result is that the gap between the distortions of embedding of a finite metric ρ of sizen intol 2 versus intol dom1 (ρ) is at most\(O\left( {\sqrt {\log n} } \right)\), and that this bound is essentially tight. A significant part of the paper is devoted to proving lower bounds on distortion of such embeddings. We also discuss some general properties and concrete examples.

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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

  2. CS Department, Haifa University, 31905, Haifa, Israel

    Yuri Rabinovich

Authors
  1. Jiří Matoušek
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  2. Yuri Rabinovich
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Corresponding author

Correspondence to Jiří Matoušek.

Additional information

Research by J. M. supported by Charles University grants No. 158/99 and 159/99. Part of the work by Y. R. was done during his visit at the Charles University in Prague partially supported by these grants, by the grant GAČR 201/99/0242, and by Haifa University grant for Promotion of Research.

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Matoušek, J., Rabinovich, Y. On dominatedl 1 metrics. Isr. J. Math. 123, 285–301 (2001). https://doi.org/10.1007/BF02784132

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  • DOI: https://doi.org/10.1007/BF02784132

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