Related papers: The local geometry of maps with c-convex potential…
We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class $W^{2,p}_{loc}$ for any $p \geq 1$ outside of a closed singular set of measure zero. We…
We construct homogeneous optimal transport maps for the quadratic cost between convex cones with homogeneous, possibly degenerate, densities when the cones satisfy an obliqueness condition. The existence of such maps plays a central role in…
A cornerstone of convex analysis, established by Rockafellar in 1966, asserts that a set has a potential if and only if it is cyclically monotone. This characterization was generalized to hold for any real-valued cost function $c$ and lies…
We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are…
In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this paper, the Jacobian determinant of a gradient map is shown to be log-concave with respect to a…
We study solutions to the multi-marginal Monge-Kantorovich problem which are concentrated on several graphs over the first marginal. We first present two general conditions on the cost function which ensure, respectively, that any solution…
Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally…
The Ma-Trudinger-Wang curvature --- or cross-curvature --- is an object arising in the regularity theory of optimal transportation. If the transportation cost is derived from a Hamiltonian action, we show its cross-curvature can be…
Weak optimal transport generalizes the classical theory of optimal transportation to nonlinear cost functions and covers a range of problems that lie beyond the traditional theory - including entropic transport, martingale transport, and…
We consider the Monge problem of optimal transport between a compactly supported source measure and a target probability measure with unbounded support. We consider the convergence of optimal maps and potential functions when the target…
We prove a nonsmooth implicit function theorem applicable to the zero set of the difference of convex functions. This theorem is explicit and global: it gives a formula representing this zero set as a difference of convex functions which…
We give natural topological conditions on the support of the target measure under which solutions to the optimal transport problem with cost function satisfying the (weak) Ma, Trudinger, and Wang condition cannot have any isolated singular…
We consider Monge-Kantorovich optimal transport problems on $\mathbb{R}^d$, $d\ge 1$, with a convex cost function given by the cumulant generating function of a probability measure. Examples include the Wasserstein-2 transport whose cost…
We show that in any complete metric space the probability measures $\mu$ with compact and connected support are the ones having the property that the optimal tranportation distance to any other probability measure $\nu$ living on the…
Multi-marginal optimal transport plans are concentrated on c-splitting sets. It is known that, similar to the two-marginal case, c-splitting sets are c-cyclically monotone. Within a suitable framework, the converse implication was very…
We present a systematic study of conditional triangular transport maps in function spaces from the perspective of optimal transportation and with a view towards amortized Bayesian inference. More specifically, we develop a theory of…
We prove that $c$-cyclically monotone transport plans $\pi$ optimize the Monge-Kantorovich transportation problem under an additional measurability condition. This measurability condition is always satisfied for finitely valued, lower…
Consider two bounded domains $\Omega$ and $\Lambda$ in $\mathbb{R}^{2}$, and two sufficiently regular probability measures $\mu$ and $\nu$ supported on them. By Brenier's theorem, there exists a unique transportation map $T$ satisfying…
We study a form of optimal transportation surplus functions which arise in hedonic pricing models. We derive a formula for the Ma-Trudinger-Wang curvature of these functions, yielding necessary and sufficient conditions for them to satisfy…
In this short note, we show that given a cost function $c$, any coupling $\pi$ of two probability measures where the second is a discrete measure can be associated to a certain bipartite graph containing a perfect matching, based on the…