Nonlinear Conservation Laws and Applications pp 217-233 | Cite as

# The Monge Problem in Geodesic Spaces

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## Abstract

We address the Monge problem in metric spaces with a geodesic distance: (*X*, *d*) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.

## Keywords

Polish Space Transport Problem Optimal Transportation Distance Cost Geodesic Space
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