$c$-cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new approach, and we show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.

It is proved that c-cyclical monotonicity is a sufficient condition for optimality in the multi-marginal optimal transport problem with Coulomb repulsive cost. The notion of c-splitting set and some mild regularity property are the tools. The result may be extended to Coulomb like costs.

We study a multimarginal optimal transportation problem in one dimension. For a symmetric, repulsive cost function, we show that, given a minimizing transport plan, its symmetrization is induced by a cyclical map, and that the symmetric optimal plan is unique. The class of costs that we consider includes, in particular, the Coulomb cost, whose optimal transport problem is strictly related to the strong interaction limit of Density Functional Theory. In this last setting, our result justifies some qualitative properties of the potentials observed in numerical experiments.

We exhibit solutions of Monge–Kantorovich mass transportation problems with constraints on the support of the feasible transportation plans and additional capacity restrictions. The Hoeffding–Fréchet inequalities are extended for bivariate distribution functions having fixed marginal distributions and satisfying additional constraints. Sharp bounds for different probabilistic functionals (e.g. Lp-distances, covariances, etc.) are given when the family of joint distribution functions has prescribed marginal distributions, satisfies restrictions on the support, and is bounded from above, or below, by other distributions.