# pr.probability – Duality of Monge-Kantorovich with a cost function \$ {0,1 } \$

Consider the usual transport problem of Monge-Kantorovich where $$X$$ and $$Y$$ are Polish spaces, $$mu$$ and $$nu$$ are probability measures on $$X$$ and $$Y$$, and $$c: X times Y to mathbb {R} ^ + cup {+ infty }$$ is a lower semi-continuous cost function. The theorem of Kantorovich's duality states that the cost of transportation between $$mu$$ and $$nu$$ is equal to the supremum of $$int_X varphi ~ d mu + int_Y psi ~ d nu$$ mostly $$L_1$$ the functions $$varphi (x)$$ and $$psi (y)$$ such as $$varphi (x) + psi (y) leq c (x, y)$$ for almost everyone $$x in X$$ and $$y in Y$$.

My question is: if $$c (x, y) in {0,1 }$$ for everyone $$x$$ and $$y$$Does this result in a solution (or solution "almost exists") where $$varphi (x)$$ and $$psi (x)$$ only take values ​​as a whole $${- 1,0,1 }$$? Finite dimensional experiments with linear programs suggest that the answer is "yes" but I can not say whether they extend to the general framework.