On Dominated L 1 Metrics

We introduce and study a class l dom 1 () of l 1-embeddable metrics corresponding to a given metric. This class is deened as the set of all convex combinations of-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 on n points with O(log n) distortion. Our main result is that the gap between the distortions of embedding of a nite metric of size n into l 2 versus into l dom 1 () is at most O(p log n), and that this bound is essentially tight. A signiicant part of the paper is devoted to proving lower bounds on distortion of such embeddings. We also discuss some general properties and concrete examples.