In optimal transport, one of the fundamentals of the theory is that the problem defining the Wasserstein/Kantorovich metric admits a dual problem. Namely that $$ \inf_{\pi \in \Pi(\mu,\nu)} \int_{X \times Y} c(x,y) d\pi = \sup_{\varphi \in C(X)} \int_X \varphi(x) + \int_Y \varphi^c(y) $$ The Prokhorov-Levy metric, defined as $$ \rho(\mu,\nu) = \inf_{\varepsilon >0 } \{ \mu(A) \leq \nu(A^\varepsilon) + \varepsilon : \forall A \in \mathcal{B}(X) \} $$ where $A^\varepsilon = \bigcup_{a \in A} B(a,\varepsilon)$. This metric also metrizes weak* convergence in $\mathcal{P}(X)$, and is also defined by an optimization problem. Is there any research into a dual problem for this? It seems hard to find any modern references on this metric at all.
I tried writing the objective in terms of $\varepsilon +$ indicators of the constraints, but attempting to apply some sort of Fenchel duality produces a conjugate that depends on the primal (I can detail this if there is interest, it doesn't seem to work).