Existence and uniqueness of optimal maps on Alexandrov spaces

Bertrand, Jérôme

doi:10.1016/j.aim.2008.06.008

Cited by 33 publications

(35 citation statements)

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“…Our discussion closely follows McCann's [22] concerning Riemannian manifolds. See [9] for the case of Euclidean spaces and [8] for Alexandrov spaces. We also refer to [2,32,41,42] for background information and further developments.…”

Section: Optimal Transport Via C-concave Functionsmentioning

confidence: 99%

Finsler interpolation inequalities

2009

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We extend Cordero-Erausquin et al.'s Riemannian Borell-Brascamp-Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani's curvature-dimension condition and a certain lower Ricci curvature bound. We also prove a new volume comparison theorem for Finsler manifolds which is of independent interest. Mathematics Subject Classification IntroductionOptimal transport theory is making rapid and breathtaking progress in recent years as it finds a large number of connections with various fields. Villani's massive lecture notes [42] provide the global picture, and a great deal of new insight as well. One of the milestones of the theory is a Riemannian Borell-Brascamp-Lieb inequality due to Cordero-Erausquin, McCann and Schmuckenschläger [11] that we will extend to Finsler manifolds. Their work offers deep inspiration as well as a technical breakthrough in the investigation of optimal transport in curved spaces. Furthermore, the Riemannian Borell-Brascamp-Lieb inequality has many meaningful applications from its ancestors (the Prékopa-Leindler inequality and the Brunn-Minkowski inequality) to the attractive curvature-dimension condition. We will also extend them to Finsler manifolds.Our generalization is twofold. We consider a Finsler manifold where one naturally encounters a nonsymmetric distance function, and we equip it with an arbitrary measure. Then the main difficulty arises from the lack of a good notion of the Hessian, and we always have to

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Section: Optimal Transport Via C-concave Functionsmentioning

confidence: 99%

Finsler interpolation inequalities

2009

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“…Unfortunately, this cannot be satisfied in more exotic topologies such as the k-holed torus (k ≥ 1), where uniqueness remains a tantalizing open question. 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 [2,5,50] and Alexandrov [11] spaces, and the mechanical actions arising from Tonelli Lagrangians [9,43]. The proof that the subtwist condition is sufficient for uniqueness relies on progress in Birkhoff's problem of characterizing extremal doubly stochastic measures on the square.…”

Section: Introductionmentioning

confidence: 99%

Optimal transportation, topology and uniqueness

2011

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The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes Communicated by N. Trudinger. Formerly titled Extremal doubly stochastic measures and optimal transportation.It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and Pierre-Andre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International Research Station (2003) containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff (Lattice Theory. Revised Edition, 1948): give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure γ outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph/antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.

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“…Notice that a different proof for the Brenier theorem in Alexandrov spaces was already given by Bertrand in [5]. Brenier theorem has been recently established by Gigli [8] in non-branching spaces with Ricci-curvature bounded from below.…”

Section: 2mentioning

confidence: 99%

“…For the quadratic cost in the Euclidean setting it was proved independently by Brenier [6] and Smith and Knott [16] that there exists a unique optimal map T , given by the gradient of a convex function, provided that µ is absolutely continuous with respect to the Lebesgue measure. This result was generalized to Riemannian manifolds by McCann [13], to Alexandrov spaces by Bertrand [5] to the Heisenberg group by Ambrosio and Rigot [4] and, very recently, to non-branching metric spaces with Ricci curvature bounded from below (in the sense of Lott, Sturm and Villani) by Gigli [8]. Notice that in all these results a reference measure m (Lebesgue measure, Riemannian volume, Haar measure, etc.)…”

Section: Introductionmentioning

confidence: 99%