 The minimum of the Kantorovich problem is equal to

where the supremum runs over all pairs of bounded and continuous functions formula_10 and formula_11 such that

For formula_13, let formula_14 denote the collection of probability measures on formula_15 that have finite "formula_16"-th moment.Let formula_17 and let formula_18, where formula_19 is a convex function.The proof of this solution appears in.