Classification of Vertex-Transitive Zonotopes

Abstract

We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a $$\Gamma $$ Γ -permutahedron for some finite reflection group $$\Gamma \subset {{\,\mathrm{O}\,}}(\mathbb {R}^d)$$ Γ ⊂ O ( 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.

The edge-transitive polytopes that are not vertex-transitive

“…The relevant result then reads Theorem 2.11 (Corollary 4.6. in [10]). If P has only centrally symmetric 2-dimensional faces (that is, it is a zonotope), has all vertices on a common sphere and all edges of the same length, then P is a Γ-permutahedron.…”

“…We classify the inscribed bipartite polytopes, that is, those with coinciding radii r 1 = r 2 . This case is made especially easy by a classification result from [10]. We need the following definition:…”

“…(6, 10, 6) 4/30 ̸ < (ϵ 3 + ϵ 5 ) + ϵ 3 < 2/30 (6, 10, 10) 8/30 ̸ < (ϵ 3 + ϵ 5 ) + ϵ 5 < 2/30 (6,10,4,4) 14/30 ̸ < (ϵ…”

Isoperimetric problems for zonotopes

“…Zonotopes have been in the focus of research since the middle of the 20th century, earning a separate chapter in the famous book [16] of Coxeter in 1943 and a place in the problem collection [44] of Shephard. They are connected to various branches of mathematics, as an example, we may mention the fact that their face lattices correspond to the combinatorial classes of central hyperplane arrangements in $$\mathbb {R}^{d+1}$$ [25, 47]. They play a central role in the theory of projections of convex polytopes, both being the projections of affine cubes (see, e.g., [37]), and, by Cauchy's projection formula [21], being the projection bodies of convex polytopes in $$\mathbb {R}^d$$ (see also [31]).…”