01330nas a2200157   4500008004100000022001400041245005900055210005500114260000800169300001400177490000800191520088200199100002301081700002201104856004601126       2013                        eng d  a1432-091600aThe Monge Problem for Distance Cost in Geodesic Spaces0 aMonge Problem for Distance Cost in Geodesic Spaces  cMar  a615–6730 v3183 a<p>We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dLis 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 π 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.</p>1 aBianchini, Stefano1 aCavalletti, Fabio  uhttps://doi.org/10.1007/s00220-013-1663-8