Before revision:  We consider the problem of transporting \mu to \nu in a way that minimizes a cost c: X \times Y }\to  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 \mathbb{R of the form c(x,y)= \Psi(x-y) for some convex function \Psi.