Weak Optimal Transport: When is the Dual Potential Convex?

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 applications in mechanism design. As in the classical case, the weak transport problem can also be written as a dual maximization problem over a pair of conjugate potentials. We identify sharp monotonicity conditions on the cost under which the dual problem can be restricted to convex potentials. This framework unifies several known results from the literature, including barycentric transport, martingale Benamou-Brenier, the multiple-good monopolist problem, Strassen's theorem, stochastic order projections and the classical Brenier theorem.