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.

我们给出了顶点传递带状昆虫的完整分类。我们证明了一个点传递zonotope是\（\伽玛\） -permutahedron对于一些有限反射组\（\伽玛\子集{{\，\ mathrm {Ó} \，}}（\ mathbb {R} ^ d ）\）。对于所有顶点都在同一球体上且所有边的长度相同的区域，也是如此。然后根据有限反射组的分类对它们进行分类。我们证明根系统可以被描述为向量的那些中心对称集合，对于这些集合，所有与恰好包含向量一半的半空间交集都是一致的。我们为中心对称集成为根系统提供了进一步的充分条件。