Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 348 / 3 / b / Solution

Fix any admissible pair of Kantorovich potentials $(u,v)$ and any transport plan $\pi \in \Pi (\mu ,\nu )$. Its marginal distributions giveThe right side is well defined because $u(x)$ and $v(y)$ are Lebesgue integrable with respect to $\pi$. Integrating their pointwise feasibility inequality givesSince this holds for every feasible pair and every transport plan,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.