Mass transport and uniform rectifiability

TL;DR

This paper links the geometric concept of uniform rectifiability to the Wasserstein distance from optimal mass transport, providing a new characterization through a localization theorem for $W_2$.

Contribution

It introduces a novel characterization of uniformly rectifiable sets using Wasserstein distance and proves a localization theorem for $W_2$ in the context of probability measures.

Findings

Characterization of uniformly rectifiable sets via Wasserstein distance.

A localization theorem for $W_2$ distance involving probability measures and radial bump functions.

Establishment of bounds for $W_2$ under certain measure conditions.

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