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Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)
, 2010
"... The variant A3w of Ma, Trudinger and Wang’s condition for regularity of optimal transportation maps is implied by the nonnegativity of a pseudoRiemannian curvature — which we call crosscurvature — induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) c ..."
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Cited by 31 (8 self)
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The variant A3w of Ma, Trudinger and Wang’s condition for regularity of optimal transportation maps is implied by the nonnegativity of a pseudoRiemannian curvature — which we call crosscurvature — induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) crosscurvature nonnegativity is preserved for products of two manifolds; (2) both A3w and crosscurvature nonnegativity are inherited by Riemannian submersions, as is domain convexity for the exponential maps; and (3) the ndimensional round sphere satisfies crosscurvature nonnegativity. From these results, a large new class of Riemannian manifolds satisfying crosscurvature nonnegativity (thus A3w) is obtained, including many whose sectional curvature is far from constant. All known obstructions to the regularity of optimal maps are absent from these manifolds, making them a class for which it is natural to conjecture that regularity holds. This conjecture is confirmed for certain Riemannian submersions of the sphere such as the complex projective spaces CPn.
Nearly round spheres look convex
 of Progress in Mathematics
"... Abstract. We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 topology, has uniformly convex injectivity domains — so M appears uniformly convex in any exponential chart. The proof is based on the Ma–Trudinger–Wang nonlocal curvature tensor, which originates from t ..."
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Cited by 29 (10 self)
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Abstract. We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 topology, has uniformly convex injectivity domains — so M appears uniformly convex in any exponential chart. The proof is based on the Ma–Trudinger–Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport.
Regularity of optimal transport maps on multiple products of spheres
, 2010
"... This article addresses regularity of optimal transport maps for cost=“squared distance” on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be nonnegatively crosscurved [KM2]. Under boundedness and nonvanishi ..."
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Cited by 23 (11 self)
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This article addresses regularity of optimal transport maps for cost=“squared distance” on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be nonnegatively crosscurved [KM2]. Under boundedness and nonvanishing assumptions on the transfered source and target densities we show that optimal maps stay away from the cutlocus (where the cost exhibits singularity), and obtain injectivity and continuity of optimal maps. This together with the result of Liu, Trudinger and Wang [LTW] also implies higher regularity (C 1,α /C ∞) of optimal maps for more smooth (C α /C ∞ ) densities. These are the first global regularity results which we are aware of concerning optimal maps on nonflat Riemannian manifolds which possess some vanishing sectional curvatures. Moreover, such product manifolds have potential relevance in statistics (see [S]) and in statistical mechanics (where the state of a system consisting of many spins is classically modeled by a point in the phase space obtained by taking many products of spheres). For the proof we apply and extend the method developed in [FKM1], where we showed injectivity and continuity of optimal maps on domains in
Uniqueness and Monge solutions in the multimarginal optimal transportation problem
 SIAM Journal on Mathematical Analysis
"... Abstract. We study a multimarginal optimal transportation problem. Under certain conditions on the cost function and the first marginal, we prove that the solution to the relaxed, Kantorovich version of the problem induces a solution to the Monge problem and that the solutions to both problems are u ..."
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Cited by 21 (3 self)
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Abstract. We study a multimarginal optimal transportation problem. Under certain conditions on the cost function and the first marginal, we prove that the solution to the relaxed, Kantorovich version of the problem induces a solution to the Monge problem and that the solutions to both problems are unique. We also exhibit several examples of cost functions under which our conditions are satisfied, including one arising in a hedonic pricing model in mathematical economics.
Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds
 Int. Math. Res. Not. IMRN 2008, Art. ID rnn120
"... Abstract. Counterexamples to continuity of optimal transportation on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, & Wang is not guaranteed by positivity of sectional curvature. 1. ..."
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Cited by 21 (4 self)
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Abstract. Counterexamples to continuity of optimal transportation on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, & Wang is not guaranteed by positivity of sectional curvature. 1.
On the MaTrudingerWang Curvature on Surfaces
"... We investigate the properties of the Ma–Trudinger–Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma–Trudinger–Wang condition is stable under C^4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positi ..."
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Cited by 21 (7 self)
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We investigate the properties of the Ma–Trudinger–Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma–Trudinger–Wang condition is stable under C^4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the Ma–Trudinger–Wang condition. As a corollary of our results, optimal transport maps on a “sufficiently flat” ellipsoid are in general nonsmooth.
Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds
 Tohoku Math. J
"... Abstract. In this paper we continue the investigation of the regularity of optimal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma–Trudinger–Wang and the geometry of the cut locus. We derive some sufficient and some necessary conditions to ensure that the opt ..."
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Cited by 20 (12 self)
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Abstract. In this paper we continue the investigation of the regularity of optimal transport maps on Riemannian manifolds, in relation with the geometric conditions of Ma–Trudinger–Wang and the geometry of the cut locus. We derive some sufficient and some necessary conditions to ensure that the optimal transport map is always continuous. In dimension two, we can sharpen our result into a necessary and sufficient condition. We also provide some sufficient conditions for regularity, and review existing results. 1.
Continuity and injectivity of optimal maps for nonnegatively crosscurved costs
, 2009
"... Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x, y). If the source density f + (x) is bounded away from zero and infinity in an open region U ′ ⊂ R n, and ..."
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Cited by 19 (9 self)
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Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x, y). If the source density f + (x) is bounded away from zero and infinity in an open region U ′ ⊂ R n, and the target density f − (y) is bounded away from zero and infinity on its support V ⊂ R n, which is strongly cconvex with respect to U ′, and the transportation cost c is nonnegatively crosscurved, we deduce continuity and injectivity of the optimal map inside U ′ (so that the associated potential u belongs to C 1 (U ′)). This result provides a crucial step in the low/interior regularity setting: in a subsequent paper [15], we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. The present paper also provides an argument required by Figalli and Loeper to conclude in two dimensions continuity of optimal maps under the weaker (in fact, necessary) hypothesis (A3w) [17]. In higher dimensions, if the densities f ± are Hölder continuous, our result permits continuous differentiability of the map inside U ′ (in fact, C 2,α