A note on some non-local variational problems

We study two non-local variational problems that are characterized by the presence of a Riesz-like repulsive term that competes with an attractive term. The first functional is defined on the subsets of the Euclidean space and has the fractional perimeter as an attractive term. The second functional instead is defined on non-negative integrable and uniformly bounded densities and contains an attractive term of positive-power type. For both of the functionals, we prove that balls are the unique minimizers in the appropriate volume constraint range, generalizing the results already present in the literature for more specific energies.