Let $\xi$ be an integral dominant weight of a root system $\Delta$, and let $\mathcal{O}_{\xi}$ be its orbit under the action of the Weyl group. The elements of the orbit are the vertices of an $n$-dimensional convex polytope.
Is there an efficient way to pick out all the subsets of points in $\mathcal{O}_{\xi}$ forming the faces of the convex polytope?