The Monge problem for strictly convex norms in $\mathbb{R}^d$

Champion, Thierry; Pascale, Luigi De

doi:10.4171/jems/234

Cited by 35 publications

(33 citation statements)

References 26 publications

(33 reference statements)

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“…This transport planγ is unique and it is known to be the unique optimal transport plan from µ to ν which is monotone on transport rays (see for instance [3,4,6,11]; notice that the functional C 2 could have been replaced by any functional γ → φ(x − y)dγ for a strictly convex function φ).…”

Section: Discrete Approximation Of the Ray-monotone Optimal Transportmentioning

confidence: 97%

“…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][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.…”

Section: Introductionmentioning

confidence: 99%

“…Not only, lower bounds would also be useful to prove some density results on the transport set: the techniques developed by Champion and De Pascale in [11] would allow in such a case to derive the existence of an optimal transport map in Monge's problem. To mention, on the other hand, applications of upper bounds and L p estimates on the transport density, we refer to [10], where these results are used to prove well-posedness for congestion problems and continuous Wardrop equilibria.…”

Section: Introductionmentioning

confidence: 99%

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Absolute continuity and summability of transport densities: simpler proofs and new estimates

Santambrogio

2009

Calc. Var.

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Section: Discrete Approximation Of the Ray-monotone Optimal Transportmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Absolute continuity and summability of transport densities: simpler proofs and new estimates

Santambrogio

2009

Calc. Var.

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“…In [2] the thesis is instead gained for a particular norm, crystalline, which is neither strictly convex, nor symmetric. The problem with merely strictly convex norms has been solved also in [9], with a different technique, focusing on convex bounded domains.…”

Section: Introductionmentioning

confidence: 99%

A proof of Sudakov theorem with strictly convex norms

Caravenna

2010

Math. Z.

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he paper establishes a solution to the Monge problem in ℝ for a possibly asymmetric norm cost function and absolutely continuous initial measures, under the assumption that the unit ball is strictly convex-but not necessarily differentiable nor uniformly convex. The proof follows the strategy initially proposed by Sudakov in 1976, found to be incomplete in 2000; the missing step is fixed in the above case adapting a disintegration technique introduced for a variational problem. By strict convexity, mass moves along rays, and we also investigate the divergence of the vector field of rays

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“…Prior to that, the existence was also obtained in [16] under some assumptions, and obtained in [30], with a gap fixed in [1]. See also [3,9,10] for the existence of optimal mappings when the norm (1.1) is replaced by a more general norm in the Euclidean space. The proofs in [8,32] are very similar: both use the approximation |x − y| 1+ε → |x − y| (ε → 0).…”

Section: Introductionmentioning

confidence: 99%

Regularity in Monge's mass transfer problem

Santambrogio

Wang

2014

Journal de Mathématiques Pures et Appliquées

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Abstract. In this paper, we study the regularity of optimal mappings in Monge's mass transfer problem. Using the approximation to Monge's cost function c(x, y) = |x − y| through the costs c ε (x, y) = ε 2 + |x − y| 2 , we consider the optimal mappings T ε for these costs, and we prove that the eigenvalues of the Jacobian matrix DT ε , which are all positive, are locally uniformly bounded. By an example we prove that T ε is in general not uniformly Lipschitz continuous as ε → 0, even if the mass distributions are positive and smooth, and the domains are c-convex. IntroductionThe Monge mass transfer problem consists in finding an optimal mapping from one mass distribution to another one such that the total cost is minimized among all measure preserving mappings. This problem was first proposed by Monge [27] and has been studied by many authors in the last two hundred years: among the main achievements in the 20th century we cite [21] and [16].In Monge's problem, the cost of moving a mass from point x to point y is proportional to the distance |x − y|, namely the cost function is given byThis is a natural cost function. In the last two decades, due to a range of applications, the optimal transportation for more general cost functions has been a subject of extensive studies. In order to present the framework more precisely, let Ω and Ω * be two bounded domains in the Euclidean space R n , and let f and g be two densities in Ω and Ω * respectively, satisfying the mass balance conditionLet c be a smooth cost function defined on Ω × Ω * .2010 Mathematics Subject Classification. 35J60, 35B65, 49Q20.

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The Monge problem for strictly convex norms in $\mathbb{R}^d$

Cited by 35 publications

References 26 publications

Absolute continuity and summability of transport densities: simpler proofs and new estimates

Absolute continuity and summability of transport densities: simpler proofs and new estimates

A proof of Sudakov theorem with strictly convex norms

Regularity in Monge's mass transfer problem

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