Cyclically monotone non-optimal N-marginal transport plans and Smirnov-type decompositions for N-flows

Abstract

In the setting of optimal transport with N ≥ 2 marginals, a necessary condition for transport plans to be optimal is that they are c-cyclically monotone. For N = 2 there exist several proofs that in very general settings c-cyclical monotonicity is also sufficient for optimality, while for N ≥ 3 this is only known under strong conditions on c. Here we give a counterexample which shows that c-cylclical monotonicity is in general not sufficient for optimality if N ≥ 3. Comparison with the N = 2 case shows how the main proof strategies valid for the case N = 2 might fail for N ≥ 3. We leave open the question of what is the optimal condition on c under which c-cyclical monotonicity is sufficient for optimality. The new concept of an N-flow seems to be helpful for understanding the counterexample: our construction is based on the absence of finite-support closed N-flows in the set where our counterexample cost c is finite. To follow this idea we formulate a Smirnov-type decomposition for N-flows.