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 . To ensure that the solution to this problem is smooth, it is necessary to make several assumptions about the structure of the domains and the cost function. In particular, Ma, Trudinger, and Wang established regularity estimates when the domains are strongly form c(x,y)= \Psi(x-y) for some convex function \Psi .