Abstract. Let M andM be n-dimensional manifolds equipped with suitable Borel probability measures ρ andρ. For subdomains M andM of R n , Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C 4 (M ×M) to guarantee smoothness of the optimal map pushing ρ forward toρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M ×M. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry.Using the pseudo-Riemannian structure, we extend Ma, Trudinger and Wang's conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principle characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and M of R n . This maximum principle plays a key role in Loeper's Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered.

Abstract. A principal wishes to transact business with a multidimensional distribution of agents whose preferences are known only in the aggregate. Assuming a twist (= generalized SpenceMirrlees single-crossing) hypothesis and that agents can choose only pure strategies, we identify a structural condition on the preference b(x, y) of agent type x for product type y -and on the principal's costs c(y) -which is necessary and sufficient for reducing the profit maximization problem faced by the principal to a convex program. This is a key step toward making the principal's problem theoretically and computationally tractable; in particular, it allows us to derive uniqueness and stability of the principal's optimum strategy -and similarly of the strategy maximizing the expected welfare of the agents when the principal's profitability is constrained. We call this condition non-negative cross-curvature: it is also (i) necessary and sufficient to guarantee convexity of the set of b-convex functions, (ii) invariant under reparametrization of agent and/or product types by diffeomorphisms, and (iii) a strengthening of Ma, Trudinger and Wang's necessary and sufficient condition (A3w) for continuity of the correspondence between an exogenously prescribed distribution of agents and of products. We derive the persistence of economic effects such as the desirability for a monopoly to establish prices so high they effectively exclude a positive fraction of its potential customers, in nearly the full range of non-negatively cross-curved models.

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 c-convex with respect to U ′ , and the transportation cost c satisfies the (A3) w condition of Trudinger and Wang [51], we deduce local Hölder continuity and injectivity of the optimal map inside U ′ (so that the associated potential u belongs to C 1,α loc (U ′ )). Here the exponent α > 0 depends only on the dimension and the bounds on the densities, but not on c. Our result provides a crucial step in the low/interior regularity setting: in a sequel [17], we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. Three key tools are introduced in the present paper. Namely, we first find a transformation that under (A3) w makes c-convex functions levelset convex (as was also obtained independently from us by Liu [40]). We then derive new Alexandrov type estimates for the level-set convex c-convex functions, and a topological lemma showing optimal maps do not mix interior with boundary. This topological lemma, which does not require (A3) w , is needed by Figalli and Loeper [20] to conclude continuity of optimal maps in two dimensions. 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 loc regularity of the associated potential) to be deduced from the work of Liu, Trudinger and Wang [41]. * An earlier draft of this paper was circulated under the title Continuity and injectivity of optimal maps for non-negatively cross-curved costs at arxiv.org/abs/0911.

We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures Ω ∈ P (P (M )) on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier [1] to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschläger [9] for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where Ω is not finitely supported. As applications, we prove versions of Jensen's inequality on Wasserstein space and a generalized Brunn-Minkowski inequality for a random measurable set on a Riemannian manifold.

The variant A3w of Ma, Trudinger and Wang's condition for regularity of optimal transportation maps is implied by the non-negativity of a pseudo-Riemannian curvature-which we call cross-curvature-induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature non-negativity is preserved for products of two manifolds; (2) both A3w and cross-curvature non-negativity are inherited by Riemannian submersions, as is domain convexity for the exponential maps; and (3) the n-dimensional round sphere satisfies cross-curvature non-negativity. From these results, a large new class of Riemannian manifolds satisfying cross-curvature non-negativity (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 CP n .

We develop a general condition on the cost function which is sufficient to imply a Monge solution and uniqueness results in the multi-marginal optimal transport problem. This result unifies and generalizes several results in the rather fragmented literature on multi-marginal problems. We also provide a systematic way to generate new examples from existing ones.

Given two probability measures µ and ν in "convex order" on R d , we study the profile of one-step martingale plans π on R d × R d that optimize the expected value of the modulus of their increment among all martingales having µ and ν as marginals. While there is a great deal of results for the real line (i.e., when d = 1), much less is known in the richer and more delicate higher dimensional case that we tackle in this paper. We show that many structural results can be obtained whenever a natural dual optimization problem is attained, provided the initial measure µ is absolutely continuous with respect to the Lebesgue measure. One such a property is that µ-almost every x in R d is transported by the optimal martingale plan into a probability measure π x concentrated on the extreme points of the closed convex hull of its support. This will be established in full generality in the 2-dimensional case, and also for any d ≥ 3 as long as the marginals are in "subharmonic order". In some cases, π x is supported on the vertices of a k(x)-dimensional polytope, such as when the target measure is discrete. Many of the proofs rely on a remarkable decomposition of "martingale supporting" Borel subsets of R d × R d into a collection of mutually disjoint components by means of a "convex paving" of the source space. If the martingale is optimal, then each of the components in the decomposition supports a restricted optimal martingale transport for which the dual problem is attained. These decompositions are used to obtain structural results in cases where duality is not attained. On the other hand, they can also be related to higher dimensional Nikodym sets.

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