Geodesic
Polar Root Polytopes that are Zonotopes
Séminaire lotharingien de combinatoire, Tome 73 (2015-2016) 
Paola Cellini; Mario Marietti. Polar Root Polytopes that are Zonotopes. Séminaire lotharingien de combinatoire, Tome 73 (2015-2016). http://geodesic.mathdoc.fr/item/SLC_2015-2016_73_a0/
@article{SLC_2015-2016_73_a0,
     author = {Paola Cellini and Mario Marietti},
     title = {Polar {Root} {Polytopes} that are {Zonotopes}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {2015-2016},
     volume = {73},
     url = {http://geodesic.mathdoc.fr/item/SLC_2015-2016_73_a0/}
}
TY  - JOUR
AU  - Paola Cellini
AU  - Mario Marietti
TI  - Polar Root Polytopes that are Zonotopes
JO  - Séminaire lotharingien de combinatoire
PY  - 2015-2016
VL  - 73
UR  - http://geodesic.mathdoc.fr/item/SLC_2015-2016_73_a0/
ID  - SLC_2015-2016_73_a0
ER  - 
%0 Journal Article
%A Paola Cellini
%A Mario Marietti
%T Polar Root Polytopes that are Zonotopes
%J Séminaire lotharingien de combinatoire
%D 2015-2016
%V 73
%U http://geodesic.mathdoc.fr/item/SLC_2015-2016_73_a0/
%F SLC_2015-2016_73_a0

Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

Let P\Phi be the root polytope of a finite irreducible crystallographic root system \Phi, i.e., the convex hull of all roots in \Phi. The polar of P\Phi, denoted P\Phi*, coincides with the union of the orbit of the fundamental alcove under the action of the Weyl group. In this paper, we determine which polytopes P\Phi* are zonotopes and which are not. The proof is constructive.