Volume Distortion for Subsets of Euclidean Spaces

In [Rao, SoCG 1999], it is shown that every n-point Euclidean metric with polynomial spread admits a Euclidean embedding with k-dimensional distortion bounded by O( √ log n log k), a result which is tight for constant values of k. We show that this holds without any assumption on the spread, and give an improved bound of O( √ log n(log k)). Our main result is an upper bound of O( √ log n log log n) independent of the value of k, nearly resolving the main open questions of [Dunagan-Vempala, RANDOM 2001] and [Krauthgamer-Linial-Magen, Discrete Comput. Geom. 2004]. The best previous bound was O(log n), and our bound is nearly tight, as even the 2-dimensional volume distortion of an n-vertex path is Ω( √ log n).