2022/12/02 by David Bate, Ana Lucia Garcia-Pulido, Bate, David +1
Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2212.01280
openalex publication_date 2022/12/02 · openalex created_date 2022/12/17 · openalex updated_date 2026/07/28
Let Ω⊂ ℝn be non-empty, open and proper. Consider Wb(Ω), the space of finite Borel measures on Ω equipped with the partial transportation metric introduced by Figalli and Gigli that allows the creation and destruction of mass on ∂ Ω. Equivalently, we show that Wb(Ω) is isometric to a subset of all Borel measures with the ordinary Wasserstein distance, on the one point completion of Ω equipped with the shortcut metric δ(x,y)= min\‖x-y‖, dist(x,∂ Ω)+dist(y,∂Ω)\. In this article we construct bi-Lipschitz embeddings of the set of unordered m-tuples in Wb(Ω) into Hilbert space. This generalises Almgren's bi-Lipschitz embedding theorem to the setting of optimal partial transport.