Existence of a lower bound for the distance between point masses of relative equilibria in spaces of constant curvature

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 $\mathbb{H}^{3}$.