Optimal maps in Monge's mass transport problem

Choose a cost function c(x) greater than or equal to 0 which is either strictly convex on Rd, or a strictly concave function of the distance \x\. Given two non-negative functions f, g is an element of L(1) (R(d)) with the same total mass, we assert the existence and uniqueness of a map which is measure-preserving between f and g, and minimizes the mass transport cost measured against c (x - y). An analytical proof based on the Euler-Lagrange equation of a dual problem is outlined. It assumes f,g to be compactly supported, and disjointly supported in the concave case.