Prescription of Gauss curvature using optimal mass transport

Bertrand, Jérôme

doi:10.1007/s10711-016-0147-3

Cited by 24 publications

(33 citation statements)

References 17 publications

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“…We refer to [14,12] and the references therein for more on this topic. Notice that, from the mathematical point of view, a quite interesting feature of this cost function is that it is strictly convex and bounded on its domain, hence in particular it is not continuous on R n × R n (while the non real-valued cost functions have been often considered as continuous, see for instance [23,22,29,30,33,11]).…”

Section: Introductionmentioning

confidence: 99%

Kantorovich potentials and continuity of total cost for relativistic cost functions

Bertrand

Pratelli

Puel³

2018

Journal de Mathématiques Pures et Appliquées

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In this paper we consider the optimal mass transport problem for relativistic cost functions, introduced in [12] as a generalization of the relativistic heat cost. A typical example of such a cost function is ct(x, y) = h(y−x t), h being a strictly convex function when the variable lies on a given ball, and infinite otherwise. It has been already proved that, for every t larger than some critical time T > 0, existence and uniqueness of optimal maps hold; nonetheless, the existence of a Kantorovich potential is known only under quite restrictive assumptions. Moreover, the total cost corresponding to time t has been only proved to be a decreasing rightcontinuous function of t. In this paper, we extend the existence of Kantorovich potentials to a much broader setting, and we show that the total cost is a continuous function. To obtain both results the two main crucial steps are a refined "chain lemma" and the result that, for t > T , the points moving at maximal distance are negligible for the optimal plan.

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Section: Introductionmentioning

confidence: 99%

Kantorovich potentials and continuity of total cost for relativistic cost functions

Bertrand

Pratelli

Puel³

2018

Journal de Mathématiques Pures et Appliquées

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“…Proof. We show only the direct implication, the reverse implication can be found in [3]. By assumption h = h K , where K is a bounded convex set that contains the origin in its interior, and let ρ K be the radial function of K i.e.…”

Section: Generalization To Convexity-like Constraintsmentioning

confidence: 93%

“…Remark 4.1 (Number of constraints). In numerical applications, we set ε = cδ, where c is a small constant, usually in the range (1,3]. For a fixed convex domain X of R 2 , there are O(1/ε) constraints per discrete segment and O(1/ε 2 ) such discrete segments.…”

Section: Numerical Implementationmentioning

confidence: 99%

Handling Convexity-Like Constraints in Variational Problems

Mérigot¹,

Oudet²

2014

SIAM J. Numer. Anal.

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Abstract. We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give precise estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise linear functions on a mesh of the domain and by other finite-dimensional spaces such as tensor-product splines. We show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints. Our implementation relies on proximal algorithms, and can be easily parallelized, thus making it applicable to large scale problems in dimension two and three. We illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation : 3D denoising, the principal agent problem, and optimization within the class of convex bodies.

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“…He also proved well-posedness of the dual problem and the absence of the duality gap. His result was generalized later by J. Bertrand in [4]. In particularly, Bertrand constructed a transportational solution for a couple of probability measures µ = f · σ, ν under the generalized Alexandrov-type assumption:…”

Section: )mentioning

confidence: 99%

Mass Transportation Functionals on the Sphere with Applications to the Logarithmic Minkowski Problem

Kolesnikov¹

2020

MMJ

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We study the transportation problem on the unit sphere S n−1 for symmetric probability measures and the cost function c(x, y) = log 1x,y . We calculate the variation of the corresponding Kantorovich functional K and study a naturally associated metric-measure space on S n−1 endowed with a Riemannian metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are solutions to the symmetric log-Minkowski problem and prove that K satisfies the following analog of the Gaussian transportation inequality for the uniform probability measure σ on S n−1 : 1 n Ent(ν) ≥ K(σ, ν). It is shown that there exists a remarkable similarity between our results and the theory of the Kähler-Einstein equation on Euclidean space. As a by-product we obtain a new proof of uniqueness of solution to the log-Minkowski problem for the uniform measure.2010 Mathematics Subject Classification. Primary: 52A40, 90C08.

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Prescription of Gauss curvature using optimal mass transport

Cited by 24 publications

References 17 publications

Kantorovich potentials and continuity of total cost for relativistic cost functions

Kantorovich potentials and continuity of total cost for relativistic cost functions

Handling Convexity-Like Constraints in Variational Problems

Mass Transportation Functionals on the Sphere with Applications to the Logarithmic Minkowski Problem

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