Kantorovich Dual Problem (for c(x,y) = d(x,y)^2 where d is a metric)

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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. [1]

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. [1]



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 . [1]