We prove that if for the curved $n$-body problem the masses are given, theminimum distance between the point masses of a specific type of relativeequilibrium solution to that problem has a universal lower bound that is notequal to zero. We furthermore prove that the set of all such relativeequilibria is compact. This class of relative equilibria includes all relativeequilibria 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$\mathbb{H}^{3}$.
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