= Solution
Fix any admissible pair of <Kantorovich potentials> $(u,v)$ and any <transport plan> $\pi\in\Pi(\mu,\nu)$. Its <marginal distributions> give
$$
\int_Xu\,d\mu+\int_Yv\,d\nu
=\int_{X\times Y}(u(x)+v(y))\,d\pi(x,y).
$$
The right side is well defined because $u(x)$ and $v(y)$ are <Lebesgue integrable> with respect to $\pi$. Integrating their pointwise feasibility inequality gives
$$
\int_Xu\,d\mu+\int_Yv\,d\nu\leq\int_{X\times Y}c(x,y)\,d\pi(x,y).
$$
Since this holds for every feasible pair and every <transport plan>,
$$
\boxed{\sup_{(u,v)\in\mathcal A_c}\left(\int u\,d\mu+\int v\,d\nu\right)\leq\inf_{\pi\in\Pi(\mu,\nu)}\mathbb K(\pi).}
$$
This proves the required inequality directly from the transport constraints, without any <convex optimization> duality theorem. Whenever the extrema are attained, the supremum and infimum can respectively be written as a maximum and minimum. Even the <Kantorovich duality theorem> is unnecessary for this direction.
Back to article page
