Before revision:  We consider the problem of transporting \mu to \nu in a way that minimizes a cost c: X \times Y \to R of the  form c(x,y)= \Psi(x-y) for some convex function \Psi .
After revision:  We consider the problem of transporting \mu to \nu in a way that minimizes a cost c: X \times Y \to R of the with respect to each other and cost function has non-negative form c(x,y)= \Psi(x-y) for some convex function \Psi .