Kantorovich Dual Problem (for where is a metric).
The case where the cost is given by is treated as a special case due to its relationship to c-concavity and convexity. 
Relationship to c-concavity and convexity
Given a function , define by . Then . Then a function is c-concave if and only if is convex and lower semicontinuous. 
Let be probabilities over and . Suppose , which implies min(KP) and suppose that gives no mass to surfaces of class . Then there exists a unique optimal
transport map from to , and it is of the form for a convex function .