Mass transport and uniform rectifiability

2011/03/08 by Xavier Tolsa, Tolsa, Xavier · 1 citation
Mathematics · #28A75 #49Q20 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1103.1543

openalex publication_date 2011/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance W₂ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance W₂ which asserts that if μ and ν are probability measures in Rⁿ, ϕ is a radial bump function smooth enough so that ∫ϕdμ\gtrsim1, and μ has a density bounded from above and from below on the support of ϕ, then W₂(ϕμ,aϕν)≤ c W₂(μ,ν), where a=∫ϕdμ/ ∫ϕ dν.

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