  For the interaction energy with repulsive-attractive potentials, we give
generic conditions which guarantee the radial symmetry of the local minimizers
in the infinite Wasserstein distance. As a consequence, we obtain the
uniqueness of local minimizers in this topology for a class of interaction
potentials. We introduce a novel notion of concavity of the interaction
potential allowing us to show certain fractal-like behavior of the local
minimizers. We provide a family of interaction potentials such that the support
of the associated local minimizers has no isolated points and any superlevel
set has no interior points.
