Existence and uniqueness of optimal maps on Alexandrov spaces

By Jérôme Bertrand. 2007

57 citations recorded by the source corpus.

The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov's curvature bounded below, Monge's transport problem for the quadratic cost admits a unique solution.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5ab210195e99e75855ec38c086876b54ca75e120

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Wasserstein Barycenters over Riemannian manifolds (2014, unknown): “This includes for example, Alexandrov spaces [3].”
Optimal maps in essentially non-branching spaces (2016, background): “s of an optimal transport map has been obtained imposing some type ofcurvature bounds from below and/ora nice behaviourof the geodesic of the space. In particular we mention the results by - Bertrand [4] for Alexandrov spaces; Key words and phrases. optimal transport; existence of maps; uniqueness o”
On a mixture of Brenier and Strassen Theorems (2018, background): “This result has then been extended to various cost functions [26] and state spaces [4, 9, 21, 23, 24, 28, 29, 54] and has had numerous applications in PDE, geometric analysis or probability theory, see [3, 6, 72, 73] and the references therein.”
Optimal transportation, topology and uniqueness (2010, background): “Our discussion is predicated on global differentiability of the cost, since a wide variety of existence and uniqueness results concerning optimal solutions to the Monge–Kantorovich problem have been established for costs with singular sets—including distances in Riemannian [28,46,77], sub-Riemannian”
Yau's gradient estimates on Alexandrov spaces (2010, background): “constant of ut on Ω′ (see [8] for this notation.); (iv) For any t >0 and almost all x ∈ Ω′, we have (5.3) lim s→0+ ut+s(x) −ut(x) t = − |∇ut(x)|2 2 . The following lemma is similar to Lemma 3.5 in [3]. Lemma 5.2. Let t >0. Assume ut is diﬀerentiable at x ∈ Ω′. Then there exists a unique point ”
Heat Flow on Alexandrov Spaces (2010, background): “If μ is absolutely continuous with respect to H n, then a minimal geodesic (μt)t∈[0,1] from μ to any ν is unique and μt is also absolutely continuous for all t ∈ (0,1) (see [5] for a more detailed characterization of μt , and [11] for the absolute continuity).”
Finsler interpolation inequalities (2009, background): “See [9] for the case of Euclidean spaces and[ 8 ]forAlexandrovspaces.Wealsoreferto[2,32,41,42]forbackgroundinformationand further developments.”
Non-branching geodesics and optimal maps in strong $$CD(K,\infty )$$CD(K,∞)-spaces (2012, result): “Most relevant in the context of this paper are the generalizations of McCann [13] for Riemannian manifolds, of Bertrand [6] for Alexandrov spaces, and the most recent results of Gigli [9] in non-branching CD(K,N) and CD(K,∞)-spaces and of Ambrosio and the first author [5] in strongly non-branching m”