Polygonal homographic orbits in spaces of constant curvature
  We prove that the geometry of the 2-dimensional $n$-body problem for spaces of constant curvature $\kappa\neq 0$, $n\geq 3$, does not allow for polygonal homographic solutions, provided that the corresponding orbits are irregular polygons of non-constant size. 
Pieter Tibboel
math.DS