  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.
