Zonotopes

A zonotope is a convex polytope that can be described as a Minkowski sum of finitely many line segments, or equivalently as a linear image of a cube. If v1, v2, ..., vm are vectors in R^n, the zonotope generated by them is Z = sum_{i=1}^m [-vi, vi] = { sum_{i=1}^m ti vi : ti ∈ [−1, 1] }. Equivalently, Z = A[−1,1]^m, where A is the n×m matrix whose columns are the vectors vi. Thus Z is a compact, convex set, and its dimension equals the rank of the set {vi}.

Basic properties include central symmetry: Z is symmetric about the origin when the generating segments are of the form [-vi, vi]. Every zonotope is a polytope, and every face of a zonotope is itself a zonotope (obtained by fixing some coefficients ti to ±1 and letting the others vary). If m segments generate Z, then Z has at most 2m facets; in general position the bound is sharp. The vertex set of Z is a subset of { sum_{i=1}^m ε_i vi : ε_i ∈ {−1, 1} }; when the vi are suitably independent, all 2^m sign combinations yield vertices.

Relation to cubes: a zonotope is precisely a linear projection of a higher-dimensional cube. Conversely, the image of a cube under any linear map is a zonotope. This makes zonotopes a natural bridge between simple product polytopes and arbitrary projections.

Examples include parallelograms (sum of two segments), cubes (sum of three pairwise orthogonal segments in 3D), and more complex shapes such as three-dimensional zonohedra. Zonotopes arise in various areas of geometry, optimization, and combinatorics.