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. The classification of these then follows from the classification of finite reflection groups. We prove 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 to be a root system.
Highlights
A polytope is the convex hull of finitely many points
While many classes of symmetric polytopes have been classified in the past, a classification of general vertex-transitive polytopes is probably infeasible: almost every finite group is isomorphic to the symmetry group of some vertex-transitive polytope [1,6]
We show that a full classification of vertex-transitive zonotopes can be achieved, and is immediately linked to the classification of finite reflection groups and their root systems
Summary
A (convex) polytope is the convex hull of finitely many points. The zonotopes form a special class of polytopes for which several equivalent definitions are known (see e.g. [13, Sect. 7.3]): Definition 1.1 A zonotope is a polytope Z ⊂ Rd that satisfies any of the following equivalent conditions:. While many classes of symmetric polytopes have been classified in the past (e.g. regular polyopes, edge- and vertex-transitive polyhedra), a classification of general vertex-transitive polytopes is probably infeasible: almost every finite group is isomorphic to the symmetry group of some vertex-transitive polytope [1,6] (the only exceptions are certain abelian groups and dicyclic groups). For a finite centrally symmetric set R ⊂ Rd \{0} consider the intersection of R with a half space that contains exactly half the elements of R (these intersections will be called semi-stars, see Definition 2.2). For root systems, such “half-sets” are known as positive roots, and it is well known that the Weyl group acts transitively on these. We prove a further sufficient condition using only the length of the sum of the vectors in the semi-stars
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