On uniqueness of an optimal solution to the Kantorovich problem with density constraints

We study optimal transportation problems with constraints on densities of transport plans. We obtain a sharp condition for the uniqueness of an optimal solution to the Kantorovich problem with density constraints, namely that the Borel measurable cost function $h(x,y)$ satisfies the following non-degeneracy condition: $h(x,y)$ can not be expressed as a sum of functions $u(x)+v(y)$ on a set of positive measure.