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 pi 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.
The Monge Problem in Geodesic Spaces / S. Bianchini, F. Cavalletti (THE IMA VOLUMES IN MATHEMATICS AND ITS APPLICATIONS). - In: Nonlinear Conservation Laws and Applications / [a cura di] A. Bressan, Gui-Qiang G. Chen, M. Lewicka, D. Wang. - [s.l] : Springer, 2011. - ISBN 9781441995537. - pp. 217-233 (( convegno Summer Program on Nonlinear Conservation Laws and Applications Proceedings : July, 13 - 31 nel 2009 [10.1007/978-1-4419-9554-4].
The Monge Problem in Geodesic Spaces
F. CavallettiUltimo
2011
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 pi 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.
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