Classification of Vertex-Transitive Zonotopes

TL;DR

This paper fully classifies vertex-transitive zonotopes, showing they are related to finite reflection groups and root systems, and provides new characterizations of these geometric structures.

Contribution

It establishes that vertex-transitive zonotopes are essentially $ ext{Gamma}$-permutahedra linked to finite reflection groups, connecting geometric and algebraic classifications.

Findings

Vertex-transitive zonotopes are $ ext{Gamma}$-permutahedra.

Homogeneous zonotopes with vertices on a sphere are classified similarly.

New characterizations of root systems are provided.

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.