On dominated l1 metrics

Jiří Matoušek, Yuri Rabinovich

Research output: Contribution to journalArticlepeer-review


We introduce and study a class ldom1 (ρ) of l1-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 into lp-spaces, such as Bourgain's embedding of an arbitrary metric ρ on n points with O(log n) distortion. Our main result is that the gap between the distortions of embedding of a finite metric ρ of size n into l2 versus into ldom1(ρ) ia at most O(√log n), 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.

Original languageEnglish
Pages (from-to)285-301
Number of pages17
JournalIsrael Journal of Mathematics
StatePublished - 2001

ASJC Scopus subject areas

  • Mathematics (all)


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