Mass Transport and Uniform Rectifiability

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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.
Original languageEnglish
Pages (from-to)478-527
JournalGeometric and Functional Analysis
Issue number2
Publication statusPublished - 1 Apr 2012


  • Optimal mass transport
  • uniform rectifiability
  • Wasserstein distance


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