Congruence and Metrical Invariants of Zonotopes

TL;DR

This paper explores the geometric properties of zonotopes, focusing on their symmetry, shape matrices, and how these relate to their congruence, rigidity, and uniqueness, providing new proofs and extending classical theorems.

Contribution

It introduces new proofs for zonotope characterizations, relates shape matrices to congruence, and extends rigidity and uniqueness results to zonotopes based on normal vectors and facet volumes.

Findings

Congruence of zonotopes determined by shape matrices.

Shape matrices encode edge and angle information.

Results extend classical theorems on polytope rigidity and uniqueness.

Abstract

Zonotopes are studied from the point of view of central symmetry and how volumes of facets and the angles between them determine a zonotope uniquely. New proofs are given for theorems of Shephard and McMullen characterizing a zonotope by the central symmetry of faces of a fixed dimension. When a zonotope is regarded as the Minkowski sum of line segments determined by the columns of a defining matrix, the product of the transpose of that matrix and the matrix acts as a shape matrix containing information about the edges of the zonotope and the angles between them. Congruence between zonotopes is determined by equality of shape matrices. This condition is used, together with volume computations for zonotopes and their facets, to obtain results about rigidity and about the uniqueness of a zonotope given arbitrary normal-vector and facet-volume data. These provide direct proofs in the case of zonotopes of more general theorems of Alexandrov on the rigidity of convex polytopes, and Minkowski on the uniqueness of convex polytopes given certain normal-vector and facet-volume data. For a zonotope, this information is encoded in the next-to-highest exterior power of the defining matrix.