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The Monge problem for distance cost in geodesic spaces. (English) Zbl 1275.49080

Authors’ abstract: We address the Monge problem in metric spaces with a geodesic distance: \((X,d)\) is a Polish space and \(d_L\) is a geodesic Borel distance which makes \((X,d_L)\) 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 \(d_L\). 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 \(d_L\)-cyclical monotonicity is not sufficient for optimality.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
53C22 Geodesics in global differential geometry
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