For compact Lie groups \(K\) coadjoint orbitopes – convex hulls \(\mathcal {\hat O}\) of coadjoint orbits \(\mathcal O\) – are considered. Methods of convex geometry are used to give a description of such orbitopes in terms of faces for \(\mathcal {\hat O} \) and their extremal points.
There is some other convex set associated to \(\mathcal O\) – the Kostant polytope \(P\), which is the convex hull of a Weyl group orbit in the Lie algebra of some maximal torus \(T \subset K\). The main result of this paper gives a bijection \(\mathcal F(P)/W \to \mathcal F (\mathcal {\hat O} )/K\) between orbits of \(K\) on the space of faces \(\mathcal F(\mathcal {\hat O})\) for \(\mathcal {\hat O}\) and the orbit space of the natural action of the Weil group \(W\) on the space of faces \(\mathcal F (P)\) for \(P\).
Also some results about the complex geometry of \(\mathcal O\) are given. In particular, the following is proved. If \(F\) is a face of \(\mathcal {\hat O}\), then the set of extremal points \(\mathrm{ext} (F) \subset \mathcal {\hat O}\) is a closed orbit of some parabolic subgroup of the complexified Lie group \(G = K^{\mathbf C}\). Conversely, if \(P \subset G\) is a parabolic subgroup, then it has a unique closed orbit \(\mathcal O ^\prime \subset \mathcal O\) and there is a face \(F\) such that \(\mathrm{ext} (F) = \mathcal O ^\prime\).