Existence and uniqueness of optimal transport maps

  • Fabio Cavalletti

    RWTH, Department of Mathematics, Templergraben 64, D-52062 Aachen, Germany
  • Martin Huesmann

    Universität Bonn, Institut für angewandte Mathematik, Endenicher Allee 60, D-53115 Bonn, Germany


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 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.

Cite this article

Fabio Cavalletti, Martin Huesmann, Existence and uniqueness of optimal transport maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 6, pp. 1367–1377

DOI 10.1016/J.ANIHPC.2014.09.006