A representation of the Kantorovich-Functional

Abstract

This paper establishes the representation of the generalizedN-dimensional Wasserstein distance (Kantorovich-Functional)

$$W_c (P_1 ,...,P_N ): = \inf \left\{ {\int_{S^N } {c(x_1 ,...,x_N )d\mu } (x_1 ,...,x_N ):\pi _i \mu = P_i ,i = 1,...,N} \right\}$$

in the form ofW _c(P ₁,...,P _N )=sup{∑ ^N _i=1 }∫sf _i dP _i . The conditions we impose onP _i ,c andf _i enable us to follow those classical lines of arguments which lead to the Kantorovich-Rubinstein Theorem: By elementary methods we show how the result for an arbitrary metric space (S, d) can be derived from the case of finiteS. We also apply this result and the techniques of its proof in order to obtain a fairly simple proof of Strassen's Theorem.