相关论文: On strict convexity and $C^1$ regularity of potent…
We are interested in the cost-convex potentials in optimal mass transport theory, and we show by direct and geometric arguments the equivalence between cost-subdifferentials and ordinary subdifferentials of cost-convex functions, under the…
We identify a condition for regularity of optimal transport maps that requires only three derivatives of the cost function, for measures given by densities that are only bounded above and below. This new condition is equivalent to the weak…
In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger…
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…
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function.…
Let $X$ and $Y$ be domains of $\mathbb{R}^n$ equipped with respective probability measures $\mu$ and $ \nu$. We consider the problem of optimal transport from $\mu$ to $\nu$ with respect to a cost function $c: X \times Y \to \mathbb{R}$. To…
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…
We study the regularity of solutions to an optimal transportation problem where the dimension of the source is larger than that of the target. We demonstrate that if the target is $c$-convex, then the source has a canonical foliation whose…
In this paper, we extend our convexity theory for $C^2$ cost functions in optimal transportation to more general generating functions, which were originally introduced by the second author to extend the framework of optimal transportation…
We prove that if $\Omega\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $\mu$ and $\bar{\mu}$ are probability measures absolutely continuous with respect to surface measure on $\partial \Omega$, with…
This paper concerns the regularity and geometry of the free boundary in the optimal partial transport problem for general cost functions. More specifically, we prove that a $C^1$ cost implies a locally Lipschitz free boundary. As an…
We study the regularity of optimal transport maps between convex domains with quadratic cost. For nondegenerate $C^{\alpha}$-densities, we prove $C^{1, 1-\varepsilon}$-regularity of the potentials up to the boundary. If in addition the…
In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of…
In this note, we extend the regularity theory for monotone measure-preserving maps, also known as optimal transports for the quadratic cost optimal transport problem, to the case when the support of the target measure is an arbitrary convex…
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…
The inverse optimal transport problem is to find the underlying cost function from the knowledge of optimal transport plans. While this amounts to solving a linear inverse problem, in this work we will be concerned with the nonlinear…
We simplify the geometric interpretation of the weak Ma-Trudinger-Wang condition for regularity in optimal transportation and provide a geometric proof of the global c-convexity of locally $c$-convex potentials when the cost function $c$ is…
The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and…
The optimal transport problem with quadratic regularization is useful when sparse couplings are desired. The density of the optimal coupling is described by two functions called potentials; equivalently, potentials can be defined as a…
Optimal maps, solutions to the optimal transportation problems, are completely determined by the corresponding c-convex potential functions. In this paper, we give simple sufficient conditions for a smooth function to be c-convex when the…