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This note establishes that a generalization of $c$-cyclical monotonicity fromthe Monge-Kantorovich problem with two marginals gives rise to a sufficientcondition for optimality also in the multi-marginal version of that problem. Toobtain the result, the cost function is assumed to be bounded by a sum ofintegrable functions. The proof rests on ideas from martingale transport.

$c$-cyclical monotonicity as a sufficient criterion for optimality in the multi-marginal Monge-Kantorovich problem