A note on the uniqueness of solutions to the transportation problem

Let mu and nu be two probability measures on the real line and c be a lower semicontinuous function on the plane. The mass transfer problem consists i n determining a measure xi with respective marginals mu and nu that minimiz es the functional integral cd xi. In this paper we show that, whenever the function c is strictly superadditive, the solution corresponding to the low er Frechet bound is the unique optimal solution. This result also holds for the discrete version of the problem.