Central configurations of nested regular polyhedra for the spatial 2n2n-body problem

We consider 2n2n masses located at the vertices of two nested regular polyhedra with the same number of vertices. Assuming that the masses in each polyhedron are equal, we prove that for each ratio of the masses of the inner and the outer polyhedra there exists a unique ratio of the length of the edges of the inner and the outer polyhedra such that the configuration is central.