  We prove that if for the curved $n$-body problem the masses are given, the
minimum distance between the point masses of a specific type of relative
equilibrium solution to that problem has a universal lower bound that is not
equal to zero. We furthermore prove that the set of all such relative
equilibria is compact. This class of relative equilibria includes all relative
equilibria of the curved $n$-body problem in $\mathbb{S}^{2}$, $\mathbb{H}^{2}$
and a significant subset of the relative equilibria for $\mathbb{S}^{3}$ and
$\mathbb{H}^{3}$.
