A note on the uniqueness of solutions to the transportation problem

Let μ and ν be two probability measures on the real line and c be a lower semicontinuous function on the plane. The mass transfer problem consists in determining a measure ξ with respective marginals μ and ν that minimizes the functional ∫ cdξ. In this paper we show that, whenever the function c is strictly superadditive, the solution corresponding to the lower Fréchet bound is the unique optimal solution. This result also holds for the discrete version of the problem.