Geometry of minimizers for the interaction energy with mildly repulsive potentials

We show that the support of any local minimizer of the interaction energy consists of isolated points whenever the interaction potential is of class C 2 and is mildly repulsive at the origin; if moreover a minimizer is global, then its support is finite. For a particular class of potentials we prove the existence of a uniform upper bound on the cardinal of the support of a global minimizer. In the one-dimensional case we give quantitative bounds