= Kantorovich duality theorem
{c}
For <probability measures> defined as <Borel measures> on <Polish spaces> and a nonnegative <sequentially lower semicontinuous> cost, the minimum cost over <transport plans> equals the supremum of $\int u\,d\mu+\int v\,d\nu$ over integrable <Kantorovich potentials> satisfying $u(x)+v(y)\leq c(x,y)$. The primal minimum is attained; a dual maximum needs additional assumptions. Compact metric spaces and a finite continuous cost suffice for attainment of both extrema.
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