The Monge Problem for distance cost in geodesic spaces

Understand

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.