A Note On The Uniqueness Of Solutions To The Transportation Problem

Let μ and v 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 v that minimizes the functional . In this paper we show that, whenever the function c is strictly superadditive, the solution corresponding to the lower Fréchet boundis the unique optimal solution. This result also holds for the discrete version of the problem.