  We prove that if for the curved $n$-body problem in $\mathbb{S}^{k-1}$,
$k\geq 3$, the masses are given, the minimum distance between the point masses
of a specific type of relative equilibrium solution that is a generalisation of
positive elliptic relative equilibria and positive elliptic-elliptic relative
equilibria has a universal lower bound that is not equal to zero.
