Before revision: The improved Fr\'echet--Hoeffding bounds are ad-hoc estimates on the Fr\'echet class of probability distributions with given marginals and prescribed values on a subset of R d . Using an optimal transport approach, we first provide an alternative derivation of the improved upper bound. To this end, we establish a dual representation of a constrained transport problem over distributions in the aforementioned Fr\'echet class and show that the improved upper Fr\'echet--Hoeffding bound belongs to the class of admissible functions for the dual optimization problem.
After revision: The improved Fr\'echet--Hoeffding bounds are ad-hoc estimates on the Fr\'echet classes of multivariate distribution functions where additional information on the joint distribution is assumed, while uncertainty in the marginals is also possible. We derive optimal transport duality results for these Fr\'echet class and show that the improved upper Fr\'echet--Hoeffding bound belongs to the class of admissible functions for the dual optimization problem.