Regularity of optimal transport maps on locally nearly spherical manifolds

Regularity of Optimal Transport Maps

In the special case “cost=squared distance” on R, the problem was solved by Caffarelli [Caf1, Caf2, Caf3, Caf4], who proved the smoothness of the map under suitable assumptions on the regularity of the densities and on the geometry of their support. However, a major open problem in the theory was the question of regularity for more general cost functions, or for the case “cost=squared distance”...

Regularity of optimal transport maps and applications

Partial Regularity for Optimal Transport Maps

We prove that, for general cost functions on R, or for the cost d/2 on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.

Regularity of optimal transport maps on multiple products of spheres

This article addresses regularity of optimal transport maps for cost=“squared distance” on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved [KM2]. Under boundedness and non-vanishing assumptions on the transfered source and target densities we show that optimal maps stay away ...

Regularity of optimal transport maps (after Ma-TrudingerWang and

In the field of optimal transportation, one important issue is the regularity of the optimal map. There are several motivations for the investigation of the smoothness of the optimal map: • It is a typical PDE/analysis question. • It is a step towards a qualitative understanding of the optimal transport map. • If it is a general phenomenon, then non-smooth situations may be treated by regulariz...