@article {Bianchini2013,
	title = {The Monge Problem for Distance Cost in Geodesic Spaces},
	journal = {Communications in Mathematical Physics},
	volume = {318},
	number = {3},
	year = {2013},
	month = {Mar},
	pages = {615{\textendash}673},
	abstract = {<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>},
	issn = {1432-0916},
	doi = {10.1007/s00220-013-1663-8},
	url = {https://doi.org/10.1007/s00220-013-1663-8},
	author = {Stefano Bianchini and Fabio Cavalletti}
}
