Existence of optimal transport maps for crystalline norms

By Luigi Ambrosio, B. Kirchheim, A. Pratelli. 2004

101 citations recorded by the source corpus.

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In Applied Mathematics

Citations in the corpus

On the sufficiency of c-cyclical monotonicity for optimality of transport plans (2008, background): “The literature about mass transport problems is extremely wide, the interested reader may look for instance at [1–5,8,15,19].”
Existence and uniqueness of optimal maps on Alexandrov spaces (2007, background): “Ambrosio, Kirchheim and Pratelli proved the existence of optimal maps for the original Monge’s problem for crystalline norms [2].”
Optimal transportation, topology and uniqueness (2010, background): “Evans and Gangbo [42] had already solved the analogous problem with the domains replaced by disjoint Lipschitz continuous probability densities, while Sudakov’s [96] earlier construction required a claim which turned out to be true only two in dimensions [4,12]; see [12,23,26] for simplifications an”
Structure of optimal martingale transport plans in general dimensions (2015, background): “(Ambrosio, Kirchheim, and Pratelli [2]) There exist a Borel set MN ⊆ [−1, 1]3 with |[−1, 1]3 − MN | = 0 and a Borel map f = ( f1, f2) : MN → [−2, 2]2 × [−2, 2]2 such that the following holds.”
The Monge Problem for Distance Cost in Geodesic Spaces (2011, background): “Pratelli in [2], and then in fully generality independently by L.”
Free boundaries in optimal transport and Monge-Ampere obstacle problems (2010, background): “This is illustrated by the following simple example, which also indicates why new hypotheses are required to ensure uniqueness. principle have been examined extensively in the context of dynamical problems since the work of Otto [65] and Jordan, Kinderlehrer and Otto [50], see also e.g. Agueh [2], A”
Absolute continuity and summability of transport densities: simpler proofs and new estimates (2009, background): “…transport, displacement convexity and Γ−convergence; [5, 13, 14, 15, 16] are the main references for transport densities; [3, 7, 9, 17, 19] are cited as papers developing similar techniques in different contests and [5, 8, 10, 11] are presented in view of possible applications of these estimates.”
Monge problem in metric measure spaces with Riemannian curvature-dimension condition (2013, unknown): “The case of a general norm as cost function on R has been solved first in the particular case of crystalline norms in [2], and then in fully generality independently by L.”
The Monge problem for strictly convex norms in Rd (2010, methodology): “2; as explained in the introduction of [2], the direction of transportation is totally determined at any point of differentiability of a Kantorovich potential u when the norm ‖ · ‖ is strictly convex, and this information is sufficient to conclude the proof of 5.”
A User’s Guide to Optimal Transport (2013, background): “Later, existence of optimal maps for the case c(x, y) := ‖x− y‖, ‖ · ‖ being any norm has been established, at increasing levels of generality, in [9], [28], [27] (containing the most general result, for any norm) and [25].”