On dominatedl 1 metrics

We introduce and study a classl ₁ ^dom (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 ₁ ^dom (ρ) is at most , 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. 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.