Maximizers of nonlocal interactions of Wasserstein type (Q6634368)

The article investigates a max-min problem involving the Wasserstein distance between sets of equal volume. The authors focus on characterizing the maximizers of a functional defined through the minimization of the Wasserstein distance. The main result demonstrates that balls are the unique maximizers, achieved by combining symmetrization-by-reflection techniques with the uniqueness properties of optimal transport plans. The proof relies on the metric structure induced by the \(p\)-Wasserstein distance as well as on the homogeneity of the cost function which allows them to use scaling properties of the energy. In addition to the general result, the authors provide a quantitative refinement of the maximality result in one dimension, where they show that the deficit of maximality is controlled from below by the square of an asymmetry given as the \(L^1\)-distance between the ball and any density. The inequality is asymptotically sharp, in the sense that the exponent of the asymmetry cannot be lowered.