Seminar Analysis: Sara Daneri (University of Zurich)

Dimensional reduction of the optimal transport problem with convex norm costs

We consider the optimal transportation problem with cost functions given by generic convex norms in R^d and absolutely continuous first marginals. We show the existence of a partition of R^d into k-dimensional sets, k=0,...,d, such that every optimal transport plan can be characterized, via disintegration of measures, as a family of optimal transport plans each moving a conditional probability of the first marginal inside one of these k-dimensional sets, along the directions of an extremal k-dimensional cone of the convex norm.  Moreover, the conditional probabilities of the first marginal on these sets are absolutely continuous with respect to the k-dimensional Hausdorff measure on the k-dimensional sets on which they are concentrated, thus settling the longstanding Sudakov's problem of the existence of locally affine decompositions of R^d that reduce norm cost transportation problem to families of lower dimensional ones. Finally, due to the minimality of our partition with respect to this "dimensional reduction" property, applications to secondary cost functions obtained first minimizing with respect to a convex norm and then with respect to a finer one (e.g., a strictly convex one) will be shown. These results were obtained in collaboration with  Stefano Bianchini (SISSA, Trieste).

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