The $\infty$-Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps

Abstract

We consider the non-nonlinear optimal transportation problem of minimizing the cost functional $\mathcal{C}_\infty(\lambda) = \operatornamewithlimits{\lambda-ess\,sup}_{(x,y) \in \Omega^2} |y-x|$ in the set of probability measures on $\Omega^2$ having prescribed marginals. This corresponds to the question of characterizing the measures that realize the infinite Wasserstein distance. We establish the existence of “local” solutions and characterize this class with the aid of an adequate version of cyclical monotonicity. Moreover, under natural assumptions, we show that local solutions are induced by transport maps.