Monge problem in metric measure spaces with Riemannian curvature-dimension condition

By Fabio Cavalletti. 2013

53 citations recorded by the source corpus.

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space $(X,d,m)$ enjoying the Riemannian curvature-dimension condition $\RCD(K,N)$, with $N < \infty$. For the first marginal measure, we assume that $\mu_{0} \ll m$. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge-Kantorovich problem, attain the same minimal value. Moreover we prove a structure theorem for $d$-cyclically monotone sets: neglecting a set of zero $m$-measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ae8023602953146d20dac3753981da6de53658cf

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

New formulas for the Laplacian of distance functions and applications (2018, background): “5 can be proved following verbatim [12].”
The globalization theorem for the Curvature-Dimension condition (2016, unknown): “Also recall the following definitions, introduced in [23]: A+ := {x ∈ Tu : ∃z, w ∈ Γu(x), (z, w) / ∈ Ru}, A− := {x ∈ Tu : ∃z, w ∈ Γ u (x), (z, w) / ∈ Ru}.”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, methodology): “We present here a strategy already introduced in [14].”