  We prove that if for relative equilibrium solutions of a generalisation of
quasi-homogeneous $n$-body problems the masses and rotation are given, then the
minimum distance between the point masses of such a relative equilibrium has a
universal lower bound that is not equal to zero. We furthermore prove that the
set of such relative equilibria is compact and prove related results for
$n$-body problems in spaces of constant Gaussian curvature.
