Classification of Vertex-Transitive Zonotopes

Abstract

We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a $\Gamma$-permutahedron for some finite reflection group $\Gamma\subset\mathrm{O}(\mathbb R^d)$. The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length (which we call homogeneous zonotopes). The classification of these then follows from the classification of finite reflection groups. We proof that root systems can be characterized as those centrally symmetric sets of vectors, for which all intersections with half-spaces, that contain exactly half the vectors, are congruent. We provide a further sufficient condition for a centrally symmetric set being a root system.