  We investigate the shape of critical points for a free energy consisting of a
nonlocal perimeter plus a nonlocal repulsive term. In particular, we prove that
a volume-constrained critical point is necessarily a ball if its volume is
sufficiently small with respect to its isodiametric ratio, thus extending a
result previously known only for global minimizers. We also show that, at least
in one-dimension, there exist critical points with arbitrarily small volume and
large isodiametric ratio. This example shows that a constraint on the diameter
is, in general, necessary to establish the radial symmetry of the critical
points.
