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We establish a general condition on the cost function to obtain uniquenessand Monge solutions in the multi-marginal optimal transport problem, under theassumption that a given collection of the marginals are absolutely continuouswith respect to local coordinates. When only the first marginal is assumed tobe absolutely continuous, our condition is equivalent to the twist on splittingsets condition found in [23]. In addition, it is satisfied by the special costfunctions in our earlier work [32, 33], when absolute continuity is imposed oncertain other collections of marginals. We also present several new examples ofcost functions which violate the twist on splitting sets condition but satisfythe new condition introduced here; we therefore obtain Monge solution anduniqueness results for these cost functions, under regularity conditions on anappropriate subset of the marginals.

Monge solutions and uniqueness in multi-marginal optimal transport: weaker conditions on the cost, stronger conditions on the marginals