Existence and stability results in the L1 theory of optimal transportation

Ambrosio, Luigi; Pratelli, Aldo

doi:10.1007/978-3-540-44857-0_5

Cited by 136 publications

(214 citation statements)

References 27 publications

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“…A transport plan π is c-cyclically monotone if it is concentrated on a Borel set Γ ⊆ X × Y which is c-cyclically monotone in the sense that The connection between optimality and c-cyclical monotonicity was also studied in [GM96,AP03,Pra08,ST08]. )…”

Section: C-cyclical Monotonicitymentioning

confidence: 99%

Duality for Borel measurable cost functions

Beiglböck¹,

Schachermayer²

2011

Trans. Amer. Math. Soc.

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Abstract. We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if c : X × Y → [0, ∞) is an arbitrary Borel measurable cost function on the product of Polish spaces X, Y . In the course of the proof we show how to relate a non-optimal transport plan to the optimal transport costs via a "subsidy" function and how to identify the dual optimizer. We also provide some examples showing the limitations of the duality relations.

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Section: C-cyclical Monotonicitymentioning

confidence: 99%

Duality for Borel measurable cost functions

Beiglböck¹,

Schachermayer²

2011

Trans. Amer. Math. Soc.

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“…Optimal transport problems is by now a classical subject that still deserves attention. We refer to [2], [3], [4], [6], [21], [22], [23] and the surveys and books [1], [13], [25] and [26]. It has many applications, for example in economics (matching problems), [5], [7], [8], [9], [10], [11], [20].…”

Section: 2mentioning

confidence: 99%

Optimal Mass Transport in Thin Domains

Climent

Rossi

Volpe

2014

Advanced Nonlinear Studies

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“…If µ 0 contains no atomic points then it can be shown that C(µ 0 , µ 1 ) ′ s given by (2.5) and (2.6) coincide [1].…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for optimal mass transportation

Wolansky

2011

Calc. Var.

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The optimal mass transportation was introduced by Monge some 200 years ago and is, today, the source of large number of results in analysis, geometry and convexity. Here I investigate a new, surprising link between optimal transformations obtained by different Lagrangian actions on Riemannian manifolds. As a special case, for any pair of non-negative measures λ + , λ − of equal masswhere W p , p ≥ 1 is the Wasserstein distance and the infimum is over the set of probability measures in the ambient space.

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Existence and stability results in the L1 theory of optimal transportation

Cited by 136 publications

References 27 publications

Duality for Borel measurable cost functions

Duality for Borel measurable cost functions

Optimal Mass Transport in Thin Domains

Limit theorems for optimal mass transportation

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