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

In the setting of optimal transport with $N\ge 2$ 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 monotoncity is also sufficient for optimality, while for $N\ge 3$ 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 $N\ge 3$. Comparison with the $N=2$ case shows how the main proof strategies valid for the case $N=2$ might fail for $N\ge 3$. 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 $N$-cycles in the set where our counterexample cost $c$ is finite. To follow this idea we formulate a Smirnov-type decomposition for $N$-flows.