On the regularity of solutions of optimal transportation problems

By Grégoire Loeper. 2009

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We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [24], [30] for a priori estimates of the corresponding Monge–Ampère equation. It is expressed by a socalled cost-sectional curvature being non-negative. We show that when the cost function is the squared distance of a Riemannian manifold, the cost-sectional curvature yields the sectional curvature. As a consequence, if the manifold does not have non-negative sectional curvature everywhere, the optimal transport map cannot be continuous for arbitrary smooth positive data. The non-negativity of the cost-sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost-convex function (the proper generalization of a convex function) and any of its supporting functions. When the cost-sectional curvature is uniformly positive, we obtain that optimal maps are continuous or Hölder continuous under quite weak assumptions on the data, compared to what is needed in the Euclidean case. This case includes the quadratic cost on the round sphere.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/caff78ce11c6fa04d7e395e5eeb6fc3304ec2e08

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Regularity of optimal transport in curved geometry: The nonfocal case (2010, methodology): “This step is based on the strategy of Loeper [20], simplified by Kim and McCann [16, Appendices], further simplified and extended in the present work.”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “We do not want to write them here, but it is worth noting that these assumptions have later been proven to be sharp by Loeper in [217].”
STABILITY OF A 4TH-ORDER CURVATURE CONDITION ARISING IN OPTIMAL TRANSPORT THEORY (2008, methodology): “There is by now plenty of evidence that these conditions play a key role in the regularity theory of optimal transport [10, 12, 13, 14 , 15, 17, 19, 20].”
Hölder regularity of optimal mappings in optimal transportation (2009, methodology): “Loeper [ 17 ] showed that the condition is also necessary.”
Interior C 2,α Regularity for Potential Functions in Optimal Transportation (2009, methodology): “The C 1, α estimate was obtained in [ 12 , 13 ], with the optimal exponent <inline-graphic xmlns:xlink="http://www.”
On the local theory of prescribed Jacobian equations (2012, methodology): “Consequently we have the following extension of the Loeper maxmum principle in optimal transportation [12]”
The Monge–Ampère equation and its link to optimal transportation (2013, methodology): “6, Loeper [84] proved the following regularity result (recall the notation in Theorem 3.”
Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2 (2009, background): “Loeper [19] proved that the round sphere (S, g) satisfies T CP.”
Continuity, curvature, and the general covariance of optimal transportation (2007, methodology): “Furthermore h is independent of (a, ā), so has a (unique) extension h̃ to Ñ := (M̃ × M̃) \1 which turns out to satisfy (A2)–(A3s) on the bi-convex set Ñ ; as in [31], [34], [42], this can be verified using the alternative characterization of (A3w/s) via concavity (/ 2-uniform concavity) of the restr”
Wasserstein Barycenters over Riemannian manifolds (2014, background): “In fact, as shown in [14], this property holds only on so-called nonnegatively cross curved spaces, which are spaces satisfying a certain fourth order condition on the metric; this fourth order condition is a strenthened variant of the celebrated Ma-Trudinger-Wang condition [27, 40] arising in the r”
Partial regularity for optimal transport maps (2012, methodology): “Indeed, as shown by Ma, Trudinger, and Wang [33], and Loeper [31], in addition to suitable convexity assumptions on the support of the target density (or on the cut locus of the manifold when supp(g) = M [24]), a very strong structural condition on DOI 10.1007/s10240-014-0064-7 the cost function, th”
Regularity of optimal transport maps on multiple products of spheres (2010, methodology): “ the product structure, the manifold has both ﬂat and positively curved directions, thus making the regularity issue much more delicate. Especially, the powerful H¨older regularity estimate of Loeper [L1] (see also [Li]) as well as the a priori estimates of Ma, Trudinger and Wang [MTW], which are su”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “To conclude our discussion on regularity of optimal mappings, let us sketch Loeper’s Hölder continuity result [85].”