The L∞ optimal transport: infinite cyclical monotonicity and the existence of optimal transport maps

We study the nonlinear optimal transportation problem of minimizing the functional among transport plans with given marginals. We present some general results regarding the problem, particularly connecting "good" solutions to a suitable definition of cyclical monotonicity. We show that cyclically monotone transport plans are induced by transport maps in under relatively general assumptions on the first marginal and the cost function. With additional assumptions we are also able to prove results about continuity and uniqueness of these optimal maps.