Mass transport and uniform rectifiability

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 $R^n$, $\varphi$ 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 $W_2(\varphi \mu ,a\varphi \nu )\le c{W}_2(\mu ,\nu ),$ where $a=\int \varphi d\mu \mathrm{/}\int \varphi d\nu$.