Optimal transport and Choquet theory

We provide a novel interpretation of optimal transport problem as an optimisation of certain linear functional over the set of all Choquet representations of pairs of probability measures and reprove the Kantorovich duality formula. We also provide such formulation and a proof for multimarginal optimal transport. We also consider case of martingale optimal transport. Here we provide a novel formulation of the dual problem. We compute the set of extreme points of multidimensional probability measures in convex order. We exhibit a link to uniformly convex and uniformly smooth functions and provide a new characterisation of such functions. We introduce notion of martingale triangle inequality and prove that if it is satisfied by a cost function, then in the dual problem to martingale optimal transport one may restrict the supremum to a narrower class of pairs of functions that are equal to each other. We prove that any such function on a convex subset of Euclidean space admits extension to the whole space.