Mass transport and uniform rectifiability

Details Mass transport and uniform rectifiability Mass transport and uniform rectifiability

2011-03-08

arxiv.org/abs/1103.1543v4

Mathematics

Classical Analysis and ODEs

Minor corrections and adjustments. To appear in Geom. Funct. Anal

Scientific paper

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 $a=\int\phi d\mu/ \int\phi\,d\nu$.

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