Chenciner and Jimenez Perez showed that the range of the spectra of the angular momenta of all the rigid motions of a fixed central configuration in a general Euclidean space form a convex polytope. In this note we explain how this result follows from a general real convexity theorem of O Shea and Sjamaar in symplectic geometry. Finally, we provide a representation theoretic description of the pushforward of the normalized measure under the real moment map for Riemannian symmetric pairs.