Let PΦ​ be the root polytope of a finite irreducible
crystallographic root system Φ, i.e., the convex hull of all roots in
Φ. The polar of PΦ​, denoted PΦ∗​,
coincides with the union of the orbit of the fundamental alcove under the
action of the Weyl group. In this paper, we establishes which polytopes
PΦ∗​ are zonotopes and which are not. The proof is
constructive.Comment: 12 page