Existence of a lower bound for the distance between point masses of   relative equilibria in $\mathbb{S}^{k-1}$, $k\geq 3$
  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. 
Pieter Tibboel
math.DS