The Monge Problem for distance cost in geodesic spaces
Abstract
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dL is a geodesic Borel distance which makes (X,dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem {\pi} which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1- dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.
Cite
@article{arxiv.1103.2796,
title = {The Monge Problem for distance cost in geodesic spaces},
author = {Stefano Bianchini and Fabio Cavalletti},
journal= {arXiv preprint arXiv:1103.2796},
year = {2015}
}