  Consider a collection of particles interacting through an
attractive-repulsive potential given as a difference of power laws and
normalized so that its unique minimum occurs at unit separation. For a range of
exponents corresponding to mild repulsion and strong attraction, we show that
the minimum energy configuration is uniquely attained -- apart from
translations and rotations -- by equidistributing the particles over the
vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in
two dimensions and regular tetrahedron in three). If the attraction is not
assumed to be strong, we show these configurations are at least local energy
minimizers in the relevant $d_\infty$ metric from optimal transportation, as
are all of the other uncountably many unbalanced configurations with the same
support. We infer the existence of phase transitions. The proof is based on a
simple isodiametric variance bound which characterizes regular simplices: it
shows that among probability measures on ${\mathbf R}^n$ whose supports have at
most unit diameter, the variance around the mean is maximized precisely by
those measures which assign mass $1/(n+1)$ to each vertex of a (unit-diameter)
regular simplex.
