Existence and stability results in the L 1 theory of optimal transportation

By Luigi Ambrosio, A. Pratelli. 2003

197 citations recorded by the source corpus.

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In Applied Mathematics

Citations in the corpus

On the extremality, uniqueness and optimality of transference plans (2009, background): “Consider an example given in [2], page 135: μ = ν = L1 with the cost given by”
Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes (2004, unknown): “The following which can be proved from Theorem 2.1 and Corollary 2.1, independently of known results on the Monge-Kantorovich problem [ 1 , 3, 4, 15, 16, 21], is our main result.”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, methodology): “Starting from the original problem by Monge, and relying on the proof strategy by Sudakov [290] (which had a gap later solved by Ambrosio in [8, 11], but was meant to treat the case of an arbitrary norm), the case of different norms has been extensively studied in the last years.”
Optimal transportation on non-compact manifolds (2007, background): “However for the case where the cost is a distance, like in (1), one should cite at least the work of Sudakov [39], Evans-Gangbo [19], Feldman-McCann [23], Caffarelli-Feldman-McCann [11], Ambrosio-Pratelli [ 5 ], and Bernard-Buffoni [8].”
The globalization theorem for the Curvature-Dimension condition (2016, result): “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”
Monge's Optimal Transport Distance for Image Classification (2016, unknown): “Monge initially considered the cost function to be proportional to the distance; this actually turned out to be a very difficult case to consider [17].”
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].”
Absolute continuity and summability of transport densities: simpler proofs and new estimates (2009, methodology): “Actually, the only bibliographical references which present specific useful tools for the proofs are [1,2, 4 ,6], while we refer to [12,18,19] for the general references on transport, displacement convexity and � −convergence;[5,13–16]arethemainreferencesfortransportdensities;[3,7,9,17,19] are cited”
Optimal transport and Skorokhod embedding (2013, background): “y monotone. Evenin thecase wherec is the squaredEuclideandistancethis a nontrivial result, posed as an open question by Villani in [Vil03, Problem 2.25]. Following contributions of AmbrosioandPratelli[AP03], this problemwas resolvedbyPratelli [Pra08]andSchachermayerTeichmann [ST09] who established t”
Optimal mass transportation and Mather theory (2004, background): “nstant K, which depends only on L, such that the measure m0 is supported on the graph of a K-Lipschitz vectorﬁeld. Proof. Let us ﬁx a probability measure µ1 on M such that C1 0(µ1,µ1) = α1. Let X : M×[0,2] −→ TMbe a vectorﬁeld associated to the transport problem C2 0(µ1,µ1) by Theorem A. Note that X”
Gradient Flows: In Metric Spaces and in the Space of Probability Measures (2005, background): “In the presentation of these facts we have been following mostly [14], [70], [111], [125].”
Sharp summability for Monge Transport density via Interpolation (2004, background): “The optimality of a transport is equivalent to the cyclical monotonicity of its graph (see [2,9] to find the definition of the cyclical monotonicity and the proof of this assert).”
Characterization of optimal transport plans for the Monge-Kantorovich problem (2007, background): “eorem ﬁnally states that a set Γ ⊂ X× Y is c-monotone if and only if there is c-concave function φsuch that Γ ⊂ ∂cφ(see [8, ch.2] and [5] for further details and references). We cite the Theorem from [1], which inspired our subsequent considerations. Theorem 1. Let X,Y be polish spaces and µ,ν be Bo”
The Monge problem for strictly convex norms in Rd (2010, methodology): “Duality and sufficiency of cyclical monotonicity may be pursued in very general settings [23, 24, 25, 3, 26, 22, 8]; however for the purpose of this paper duality may be obtained more easily [1, 29].”
Optimal and better transport plans (2008, methodology): “3 Consider the task to transport points on the real line (equipped with the Lebesgue measure) from the interval [0, 1) to [1, 2) where the cost of moving one point to another is the squared distance between these points (X = [0, 1), Y = [1, 2), c(x, y) = (x − y)(2), μ = ν = λ).”
Existence and uniqueness of optimal maps on Alexandrov spaces (2007, background): “we denote this condition by µ1 = s♯µ0). Monge’s original problem has only been solved quite recently (Sudakov gave a proof ﬁrst [20] but later, part of his proof appeared to be incorrect, we refer to [3] for details). In the meantime, the importance of the mass transport problem for other cost funct”
Duality for Borel measurable cost functions (2008, unknown): “(This equivalence is not true in general, as shown by a beautiful counterexample due to Ambrosio and Pratelli, [AP03, Example 3.5].”