A minimax approach to duality for linear distributional sensitivity testing

We consider the problem of finding the maximum of $\mathbb{E}_{\nu}[f(X)]$ where $\nu$ is allowed to vary over all the probability measures on a Polish space $S$ for which $d_c(\mu,\nu)\leq \theta$, in which $d_c$ is an optimal transport distance, $f$ a real-valued function on $S$ satisfying some regularity, $\mu$ a ``baseline" measure and $\theta \geq 0$. Whereas some of the derivations of the dual version of this optimization problem rely on Fenchel duality, we impose compactness on $S$ to allow us to instead use K. Fan's minimax theorem, which does not require vector space structure. This allows one to avoid the use of vector spaces of measures, or dual variables other than the Lagrange multiplier.