  We characterize the maximizers of a functional involving the minimization of
the Wasserstein distance between equal volume sets. This functional appears as
a repulsive interaction term in some models describing biological membranes. We
combine a symmetrization-by-reflection technique with the uniqueness of optimal
transport plans to prove that balls are the only maximizers. Further, in one
dimension, we provide a sharp quantitative version of this maximality result.
