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In the setting of optimal transport with $N\ge 2$ marginals, a necessarycondition for transport plans to be optimal is that they are $c$-cyclicallymonotone. 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 acounterexample which shows that $c$-cylclical monotonicity is in general notsufficient for optimality if $N\ge 3$. Comparison with the $N=2$ case shows howthe main proof strategies valid for the case $N=2$ might fail for $N\ge 3$. Weleave 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: ourconstruction is based on the absence of finite-support $N$-cycles in the setwhere our counterexample cost $c$ is finite. To follow this idea we formulate aSmirnov-type decomposition for $N$-flows.

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