The Monge Problem for Distance Cost in Geodesic Spaces

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 π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1dimensional 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.

Isoperimetric inequality under Measure-Contraction property

“…Localization for MCP(K, N ) was, partially and in a different form, already known in 2009, see [11,Theorem 9.5], for non-branching m.m.s.. The case of essentially nonbranching m.m.s.…”

An Overview of L1 optimal transportation on metric measure spaces

“…Hence to apply the localization technique to the Lévy-Gromov isoperimetric inequality in singular spaces, structural properties of geodesics and of L 1 -optimal transportation have to be understood also in the general framework of metric measure spaces. Such a program already started in the previous work of the author with S. Bianchini [11] and of the author [18,17]. Finally with A. Mondino in [20] we obtained the general result permitting to obtained the Lévy-Gromov isoperimetric inequality.…”

“…Let ϕ : X → R be any 1-Lipschitz function. Here we present some useful results (all of them already presented in [11]) concerning the d-cyclically monotone set associated with ϕ:…”

“…For the proof of this statement (that goes beyond the scope of this note) we refer to [11], Lemma 5.3. The same comment will also apply to the next Theorem 4.7.…”

“…Now the assumption µ 0 ≪ m is sufficient to solve the Monge problem. It is worth mentioning that [11] was the first contribution where regularity of conditional measures were obtained in a purely non-smooth framework. The techniques introduced in [11] permitted also to threat such regularity issues in the infinite dimensional setting of Wiener space; see [16].…”