Related papers: Optimal and better transport plans
We consider the optimal transportation problem on a globally hyperbolic spacetime for some cost function $c_2$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is the…
The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing…
In this paper, Monge-Kantorovich problem is considered in the infinite dimension on an abstract Wiener space $(W, H,\mu)$, where $H$ is Cameron-Martin space and $\mu$ is the Gaussian measure. We study the regularity of optimal transport…
We propose a model to describe the optimal distributions of residents and services in a prescribed urban area. The cost functional takes into account the transportation costs (according to a Monge--Kantorovich-type criterion) and two…
The Gromov--Wasserstein problem is a non-convex optimization problem over the polytope of transportation plans between two probability measures supported on two spaces, each equipped with a cost function evaluating similarities between…
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…
We present a general method, based on conjugate duality, for solving a convex minimization problem without assuming unnecessary topological restrictions on the constraint set. It leads to dual equalities and characterizations of the…
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…
We establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with…
We consider the $L^\infty$-optimal mass transportation problem \[ \min_{\Pi(\mu, \nu)} \gamma-\mathrm{ess\,sup\,} c(x,y), \] for a new class of costs $c(x,y)$ for which we introduce a tentative notion of twist condition. In particular we…
This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex…
In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and…
We investigate the approximation of Monge--Kantorovich problems on general compact metric spaces, showing that optimal values, plans and maps can be effectively approximated via a fully discrete method. First we approximate optimal values…
Let (X,L) be a (semi-) polarized complex projective variety and T a real torus acting holomorphically on X with moment polytope P. Given a probability density g on P we introduce a new type of Monge-Ampere measure on X, defined for singular…
We consider the problem to transport resources/mass while abiding by constraints on the flow through constrictions along their path between specified terminal distributions. Constrictions, conceptualized as toll stations at specified…
In this paper, we want to establish some general results in the Lorentzian optimal transport theory that have well-known Riemannian counterparts. As a first result, we will provide non-trivial assumptions on the measures to ensure strong…
We study optimal transportation problems with constraints on densities of transport plans. We obtain a sharp condition for the uniqueness of an optimal solution to the Kantorovich problem with density constraints, namely that the Borel…
We introduce and study a multi-marginal optimal partial transport problem. Under a natural and sharp condition on the dominating marginals, we establish uniqueness of the optimal plan. Our strategy of proof establishes and exploits a…
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…
We study a single-period optimal transport problem on $\mathbb{R}^2$ with a covariance-type cost function $c(x,y) = (x_1-y_1)(x_2-y_2)$ and a backward martingale constraint. We show that a transport plan $\gamma$ is optimal if and only if…