The Monge Problem for Distance Cost in Geodesic Spaces

By Stefano Bianchini, Fabio Cavalletti. 2011

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We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dL is 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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ba2caeb1d17656b445904ff08ca1f37d07de528d

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Citations in the corpus

Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “The Monge-Kantorovich theory has found a wide variety of applications in pure and applied mathematics.”
Needle decompositions and isoperimetric inequalities in Finsler geometry (2015, unknown): “nd p(Mϕ) = Rϕ, where p: Tϕ −→Rϕ is the projection. We remark that ∼is not an equivalence relation of Tϕ⊔Bϕ. Consider the example ϕ(x) = |x|in Example 4.3 and observe −1 ∼0, 0 ∼1, but −1 ∼1. Following [BC] (see also [Ca1, CM1]), we shall disintegrate m|Tϕ along Tϕ/∼, via a Borel map σ: Tϕ−→Tϕ satisfy”
New formulas for the Laplacian of distance functions and applications (2018, background): “[9]) that R u is an equivalence relation over T nb u and that for any x ∈ T nb u , Ru(x) ⊂ (X, d) is isometric to a closed interval in (R, |·|).”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “Such results have their roots in previous works of BianchiniCavalletti [13] and the first author [14, 15], and will be developed in Sections 3 and 4.”
Characterization of Low Dimensional RCD*(K, N) Spaces (2015, methodology): “erence measures for (M,d g)(in the sense of Cavalletti-Mondino [12]). Moreover m g ≪ µ and µ ≪ m g hold. Proof. If (M,g)has no boundary,theﬁrstclaim for m g follows from Proposition8.1 in[12](also see[4]). Evenif (M,g,m g)is [0,l]forsomel >0orR≥0,a similar proof as in the aforementioned paper wor”
The globalization theorem for the Curvature-Dimension condition (2016, background): “tation Theory In this section we recall various results from the theory of L1 optimal-transport which are relevant to this work, and add some new information we will subsequently require. We refer to [2, 13, 19, 23, 37, 38, 47, 75] for more details. 47 7.1 Preliminaries To any 1-Lipschitz function u”
On the topology and the boundary of N–dimensional RCD(K,N) spaces (2019, background): “}. Its transpose is given by Γ−1 u= {(x,y) ∈ X × X : (y,x) ∈ Γ }. We deﬁne the transport relation Ru as: (A.3) R u:= Γu ∪Γ −1. Using that (X,d,m) is essentially non-branching, Cavalletti [Cav14] (cf. [BC13]) proved that Ru induces a partition of X (up to a subset N, with m(N) = 0) into a disjoint fa”
Monge problem in metric measure spaces with Riemannian curvature-dimension condition (2013, unknown): “Introduction Let (X, d,m) be a metric measure space verifying the Riemannian curvature dimension condition”