Two classes of stacked central configurations for the spatial 2n+12n+1-body problem: Nested regular polyhedra plus one

In this paper we consider 2n2n mass points located at the vertices of two nested regular polyhedra with the same number of vertices and the (2n+1)(2n+1)th mass located at the geometrical center of the nested regular polyhedra. We show the existence of central configurations for any given mass ratios and the size ratio of nested polyhedra.