Remark 2.8. More generally, the main theorem in  shows that the Kantorovich duality holds, for instance, for costs of form
9.2.4 Geometry of the Optimal Transport sets
Our goal in this section is to present the relation between the support of a coupling γ and optimality in the Monge–Kantorovich problem (2.1). In the following, we will just summarize key results necessary to present recent developments of optimal transportation with Coulomb and repulsive harmonic-type costs.
Roughly speaking, it is well known in optimal transport theory with 2-marginals that for a wide class of costs c, a coupling γ is optimal if, and only if, the support ...