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Mass Transport and Uniform Rectifiability

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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 μ and ν are probability measures in \({{\mathbb{R}^n}}\) , \({{\varphi}}\) is a radial bump function smooth enough so that \({{\int \varphi d \mu \gtrsim 1}}\) , and μ has a density bounded from above and from below on supp(\({{\varphi}}\)), then \({{W_2(\varphi \mu, a\varphi \nu) \leq cW_2(\mu, \nu)}}\) , where \({{a = \int \varphi d\mu/ \int \varphi d\nu}}\) .

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Correspondence to Xavier Tolsa.

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Partially supported by grants 2009SGR-000420 (Generalitat de Catalunya) and MTM-2010-16232 (Spain).

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Tolsa, X. Mass Transport and Uniform Rectifiability. Geom. Funct. Anal. 22, 478–527 (2012). https://doi.org/10.1007/s00039-012-0160-0

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