Kostant's Convexity Theorem

Kostant's convexity theorem, introduced by Bertram Kostant in 1973, states that the projection of any coadjoint orbit of a connected compact Lie group onto the dual of a Cartan subalgebra forms a convex set, a special case of a more general convexity result for symmetric spaces. It generalizes an earlier result of Schur, Horn, and Thompson for Hermitian matrices, which showed that projecting the set of self-adjoint matrices with fixed eigenvalues onto their diagonal entries produces the convex polytope whose vertices are all permutations of those eigenvalues, and it can be used to derive Lie-theoretic extensions of the Golden-Thompson inequality and the Schur-Horn theorem.