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Séminaire Lotharingien de Combinatoire, B73a (2015), 11 pp.

# Paola Cellini and Mario Marietti

# Polar Root Polytopes that are Zonotopes

**Abstract.**
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.

Received: September 10, 2014.
Revised: February 6, 2015.
Accepted: February 16, 2015.

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