Let be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided satisfies a new weak property concerning the behavior of m under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure contraction property.We also prove a stability property for Assumption 1: If satisfies Assumption 1 and , for some continuous function , then also verifies Assumption 1. Since these changes in the reference measures do not preserve any Ricci type curvature bounds, this shows that our condition is strictly weaker than measure contraction property.
@article{AIHPC_2015__32_6_1367_0, author = {Cavalletti, Fabio and Huesmann, Martin}, title = {Existence and uniqueness of optimal transport maps}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1367--1377}, publisher = {Elsevier}, volume = {32}, number = {6}, year = {2015}, doi = {10.1016/j.anihpc.2014.09.006}, mrnumber = {3425266}, zbl = {1331.49063}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.006/} }
TY - JOUR AU - Cavalletti, Fabio AU - Huesmann, Martin TI - Existence and uniqueness of optimal transport maps JO - Annales de l'I.H.P. Analyse non linéaire PY - 2015 SP - 1367 EP - 1377 VL - 32 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.006/ DO - 10.1016/j.anihpc.2014.09.006 LA - en ID - AIHPC_2015__32_6_1367_0 ER -
%0 Journal Article %A Cavalletti, Fabio %A Huesmann, Martin %T Existence and uniqueness of optimal transport maps %J Annales de l'I.H.P. Analyse non linéaire %D 2015 %P 1367-1377 %V 32 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.006/ %R 10.1016/j.anihpc.2014.09.006 %G en %F AIHPC_2015__32_6_1367_0
Cavalletti, Fabio; Huesmann, Martin. Existence and uniqueness of optimal transport maps. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, pp. 1367-1377. doi : 10.1016/j.anihpc.2014.09.006. http://www.numdam.org/articles/10.1016/j.anihpc.2014.09.006/
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