Optimal transport maps on Alexandrov spaces revisited. (English) Zbl 1503.49037
The authors study the optimal transport problem in the setting of metric measure spaces under some curvature assumptions. The space is assumed to be Alexandrov with curvature bounded from below, i.e. its curvature is defined through a comparison between the distances in geodesic triangles with distances in a corresponding model space. Alexandrov spaces have a well-defined dimension which is either an integer or is infinite; in this paper, it is assumed to be finite and equal to \(n\). In this setting, the authors study the Monge formulation of the optimal transport problem; if the source measure is purely \((n-1)\)-unrectifiable, or in other words it does not concentrate on sets of codimension one, they prove that the solution to the Kantorovich problem is unique and it is induced by a map. Note that this is the same condition which appears in similar results on Riemannian manifolds. The proof is based on the nonbranching property for geodesics in Alexandrov spaces.
Reviewer: Wojciech Górny (Warszawa)
MSC:
| 49Q22 | Optimal transportation |
| 53C45 | Global surface theory (convex surfaces à la A. D. Aleksandrov) |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 53C22 | Geodesics in global differential geometry |
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