Trivial cases for the Kantorovitch problem

Let X and Y be two compact spaces endowed with respective measures mu and n u satisfying the condition mu(X) = nu(Y). Let c be a continuous function on the product space X x Y. The mass transfer problem consists in determining a measure xi on X x Y whose marginals coincide with mu and nu, and such th at the total cost integral integral c(x, y) d xi(x, y) be minimized. We fir st show that if the cost function c is decomposable, i.e., can represented as the su,n of two continuous functions defined on X and Y, respectively, t hen every feasible measure is optimal. Conversely, wizen X is the support o f mu and Y the support of nu and when every feasible measure is optimal, we pi-eve that the cost function is decomposable.