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 μ and ν are probability measures in ℝ n, φ is a radial bump function smooth enough so that ∫ φ dμ ≳ 1, and μ has a density bounded from above and from below on supp(φ), then W 2(φμ, aφν) ≤ cW 2(μν), where a = ∫ φdμ/∫φdν. © 2012 Springer Basel AG.