On the extremality, uniqueness and optimality of transference plans

We consider the following standard problems appearing in optimal mass transportation theory: • when a transference plan is extremal, • when a transference plan is the unique transference plan concentrated on a set A, • when a transference plan is optimal. We study these three problems with a general approach: (1) choose some necessary conditions, depending on the problem we are considering; (2) find a partition into sets Bα where these necessary conditions become also sufficient; (3) show that all the transference plans are concentrated on ∪αBα. Explicit procedures are provided in the three cases above, the principal one being that the problem has an hidden structure of linear preorder with universally measurable graph. As by sides results, we study the disintegration theorem w.r.t. a family of equivalence relations, the construction of optimal potentials, a natural relation obtained from c-cyclical monotonicity. Received and Communicated by T. P. Liu on September 22, 2009.