Duality for the $L^{\infty }$ optimal transport problem

Abstract

We derive the dual of the relaxed Monge-Kantorovich optimal mass transport problem in $L^{\infty }$ in which one seeks to minimize $\mu$-$\mathrm {ess sup}_{(x,y)\in \mathbb {R}^N \times \mathbb {R}^N} c(x,y)$ over Borel probability measures $\mu$ with given marginals $P_0, P_1.$ Several formulations of the dual problem are obtained using various techniques including quasiconvex duality. We also consider weighted optimal transport in $L^{\infty }$ and we identify the form of the dual in the Lagrangian cost setting for both integral and essential supremum costs along trajectories. Finally, we prove a duality formula that relates a maximization problem which arises naturally in the $L^{\infty }$ calculus of variations with a family of optimal partial transport problems.