2011/03/14 by Fabio Cavalletti, Cavalletti, Fabio · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #Probability (math.PR) #math.AP #math.PR
paper · pdf · doi:10.48550/arxiv.1103.2797
27 pages, 1 figure; SIAM J. Math. Anal. 2012. arXiv admin note: substantial text overlap with arXiv:1103.2796, arXiv:1103.2798
openalex publication_date 2011/03/14 · arxiv created 2012/09/28 · arxiv updated 2012/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dN is a geodesic Borel distance which makes (X,dN) a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along family of geodesics. We introduce two assumptions on the transference plan π which imply that the conditional probabilities of the first marginal on each family of geodesics are continuous and that each family of geodesics is a hourglass-like set. We show that this regularity is sufficient for the construction of a transport map. We apply these results to the Monge problem in d with smooth, convex and compact obstacle obtaining the existence of an optimal map provided the first marginal is absolutely continuous w.r.t. the d-dimensional Lebesgue measure.