Mass transport and uniform rectifiability

Abstract: In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance $W_2$ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance $W_2$ which asserts that if $\mu$ and $\nu$ are probability measures in $R^n$, $\phi$ is a radial bump function smooth enough so that $\int\phi d\mu\gtrsim1$, and $\mu$ has a density bounded from above and from below on the support of \phi, then $W_2(\phi\mu,a\phi\nu)\leq c W_2(\mu,\nu),$ where $a=\int\phi d\mu/ \int\phi\,d\nu$.