## Lei Zhao (RuG)

### Convexity of Chenciner's frequency mapping and the O'Shea-Sjamaar theorem

In studying the relative equilibrium motions of the Newtonian N-body problem in Euclidean space having high enough dimensions, Chenciner has defined the frequency mapping which associates to a fixed central configuration with fixed moment inertia the ordered spectrum of its angular momentum bivector. It is conjectured and later proved by Chenciner-Jimenez that the image of the frequency mapping is a convex polytope. The proof, which is somehow indirect, goes by realizing this image between two well-studied Horn-type convex polytopes and showed that these two convex polytopes coincide. In view of its potential link with the moment maps, Chenciner-Leclerc asked if there exists a direct, conceptual proof of this convexity property of the frequency mapping. In this talk, we present such a proof by showing that this convexity property is a direct consequence of a theorem of O'Shea-Sjamaar, which is a ``real'' version of the theorem of Kirwan concerning the convexity property of the moment map. This is a joint work with Gert Heckman (Nijmegen).