Convex Central Configurations of the n-Body Problem Which are not Strictly Convex

Abstract

It is well-known that if a planar central configuration for the Newtonian 4-body problem is convex, then it must be strictly convex. In some literature, same conclusion was believed to hold for the case of five or even more bodies but rigorous treatments are absent. With the help of some numerical calculations, in this paper we provide concrete examples of central configurations which are convex but not strictly convex. Our examples include planar central configurations with five bodies and spatial central configurations with seven bodies.