Let X and Y be two compact spaces endowed with
respective measures μ and ν satisfying the condition µ(X) = v(Y). Let c be a continuous function on the product space X x Y. The mass transfer problem consists in determining a measure ξ on
X x Y whose marginals coincide with μ and ν, and such that
the total cost ∫ ∫ c(x,y)dξ(x,y) be minimized. We first
show that if the cost function c is decomposable, i.e., can be
represented as the sum of two continuous functions defined on X and
Y, respectively, then every feasible measure is optimal. Conversely,
when X is the support of μ and Y the support of ν and when
every feasible measure is optimal, we prove that the cost function is
decomposable.