We investigate which nonlocal-interaction energies have a ground state
(global minimizer). We consider this question over the space of probability
measures and establish a sharp condition for the existence of ground states. We
show that this condition is closely related to the notion of stability (i.e.
H-stability) of pairwise interaction potentials. Our approach uses the direct
method of the calculus of variations.Comment: This version is to appear in the J Stat Phy
