Moscow Mathematical Journal

We prove a conjecture made by the first named author: Given an n-body central configuration X₀ in the euclidean space E of dimension 2p, let Im ℱ be the set of decreasing real p-tuples (ν₁,ν₂,…,ν_p) such that {±iν₁,±iν₂,…,±iν_p} is the spectrum of the angular momentum of some (periodic) relative equilibrium motion of X₀ in E. Then Im ℱ is a convex polytope. The proof consists in showing that there exist two, generically (p−1)-dimensional, convex polytopes 𝒫₁ and 𝒫₂ in ℝ^p such that 𝒫₁ ⊂ Im ℱ ⊂ 𝒫₂ and that these two polytopes coincide.

𝒫₁, introduced earlier in a paper by the first author, is the set of spectra corresponding to the hermitian structures J on E which are ``adapted'' to the symmetries of the inertia matrix S₀; it is associated with Horn's problem for the sum of p×p real symmetric matrices with spectra σ₋ and σ₊ whose union is the spectrum of S₀.

𝒫₂ is the orthogonal projection onto the set of ``hermitian spectra'' of the polytope 𝒫 associated with Horn's problem for the sum of 2p×2p real symmetric matrices having each the same spectrum as S₀$.

The equality 𝒫₁=𝒫₂ follows directly from a deep combinatorial lemma by S. Fomin, W. Fulton, C.K. Li, and Y.T. Poon, which characterizes those of the sums of two 2p×2p real symmetric matrices with the same spectrum which are hermitian for some hermitian structure.

2010 Mathematics Subject Classification. 70F10, 70E45, 15A18, 15B57