On dominated l₁ metrics

We introduce and study a class l^dom₁ (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 l_p-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 l^dom₁(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.