On the regularity of solutions of optimal transportation problems

Abstract

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 [22,29] for a priori estimates of the corresponding Monge-Ampère equation. It is expressed by a so-called 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 nonnegative 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 costconvex 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. Vol (NMoreover, for C > 0, there exists a constant C ′ > 0 such thatfor all η > 0. Then, as ψ is a smooth diffeomorphism, one has Vol (N η ([y 0 , y 1 ] xm ) ∩ Ω ′ ) /Vol (N η ([y 0 , y 1 ] xm )) ≥ C(c, Ω, Ω ′ )Vol (ψ (N η ([y 0 , y 1 ] xm ) ∩ Ω ′ )) /Vol (ψ (N η ([y 0 , y 1 ] xm ))) .

The Monge–Ampère equation and its link to optimal transportation

“…By exploiting (a variant of) Theorem 4.6, Loeper [87] proved the following regularity result (recall the notation in Theorem 3.6):…”

“…Although the MTW condition seemed the right assumption to obtain regularity of optimal maps, it was only after Loeper's work [87] that people started to have a good understanding of this condition, and a more geometric insight. The idea of Loeper was the following: for the classical MongeAmpère equation, a key property to prove regularity of convex solutions is that the subdifferential of a convex function is convex, and so in particular connected.…”

“…By slightly modifying their argument, Loeper [87] showed that the connectedness of the c-subdifferential is a necessary condition for the smoothness of optimal transport (see also [114,Theorem 12.7]): Theorem 4.5. Assume that there existx ∈ X and ψ : X → R c-convex such that ∂ c ψ(x) is not connected.…”

“…In [87] he proved (a slightly weaker version of) the following result (see [114,Chapter 12] for a more general statement): Theorem 4.6. The following conditions are equivalent:…”

“…Apart from having its own interest, this general theory of Monge-Ampère type equations satisfying the MTW condition turns out to be useful also in the classical case. Indeed the MTW condition can be proven to be coordinate invariant [90,87,76]. This implies that, if u solves an equation of the form (1.2) with A satisfying the MTW condition (see (4.4)) and Φ is a smooth diffeomorphism, then u•Φ satisfies an equation of the same form with a new matrixà which still satisfies the MTW condition.…”