Zonotope

The Minkowski sum of a finite set of line segments in \mathbb{R}^d forms a type of convex polytope called a zonotope. More precisely, a zonotope Z generated by the vectors w_1,...,w_n\in\mathbb{R}^d is a translation of Z = \{a_1 w_1 + \cdots + a_n w_n | \; 0 \le a_j \le 1 \text{ for all } j \} = \mathbf{W} \, [0,1]^n , where \mathbf{W} is the d \times n matrix whose j'th column is w_j. The latter description makes it clear that a zonotope is precisely the translation of a projection of an n-dimensional cube. In the special case where w_1,...,w_n\in\mathbb{R}^d are linearly independent, the zonotope Z is a (possibly lower-dimensional) parallelotope. The facets of any zonotope are themselves zonotopes of one lower dimension. Examples of four-dimensional zonotopes include the tesseract (Minkowski sums of mutually perpendicular equal length line segments), the omnitruncated 5-cell, and the truncated 24-cell. Every permutohedron is a zonotope. ==Zonotopes and matroids==

Fix a zonotope Z generated by the vectors w_1,\dots,w_n \in \mathbb{R}^d and let \mathbf{W} be the d \times n matrix whose columns are the w_i. Then the vector matroid {\mathcal{M}} on the columns of \mathbf{W} encodes a wealth of information about Z, that is, many properties of Z are purely combinatorial in nature. For example, pairs of opposite facets of Z are naturally indexed by the cocircuits of \mathcal{M} and if we consider the oriented matroid \mathcal{M} represented by \mathbf{W}, then we obtain a bijection between facets of Z and signed cocircuits of \mathcal{M} which extends to a poset anti-isomorphism between the face lattice of Z and the covectors of \mathcal{M} ordered by component-wise extension of 0 \prec +, -. In particular, if M and N are two matrices that differ by a projective transformation then their respective zonotopes are combinatorially equivalent. The converse of the previous statement does not hold: the segment [0,2] \subset \mathbb{R} is a zonotope and is generated by both \{2\mathbf{e}_1\} and by \{\mathbf{e}_1, \mathbf{e}_1\} whose corresponding matrices, [2] and [1~1], do not differ by a projective transformation. Tilings Tiling properties of the zonotope Z are also closely related to the oriented matroid \mathcal{M} associated to it. First we consider the space-tiling property. The zonotope Z is said to tile \mathbb{R}^d if there is a set of vectors \Lambda \subset \mathbb{R}^d such that the union of all translates Z + \lambda (\lambda \in \Lambda) is \mathbb{R}^d and any two translates intersect in a (possibly empty) face of each. Such a zonotope is called a space-tiling zonotope. The following classification of space-tiling zonotopes is due to McMullen:{{cite journal == Dissections ==

Every d-dimensional zonotope generated by a finite set A of vectors can be partitioned into parallelepipeds, with one parallelepiped for each linearly independent subset of A. This yields another family of tilings associated to the zonotope Z, given by a zonotopal tiling of Z, i.e., a polyhedral complex with support Z: the union of all zonotopes in the collection is Z and any two intersect in a common (possibly empty) face of each. The Bohne-Dress Theorem states that there is a bijection between zonotopal tilings of the zonotope Z and single-element lifts of the oriented matroid \mathcal{M} associated to Z. == Volume ==

Zonotopes admit a simple analytic formula for their volume. Let Z(S) be the zonotope Z = \{a_1 w_1 + \cdots + a_n w_n | \; \forall(j) a_j\in [0,1]\} generated by a set of vectors S = \{w_1,\dots,w_n\in\mathbb{R}^d\}. Then the d-dimensional volume of Z(S) is given by :\sum_{T\subset S \; : \; |T| = d} |\det(Z(T))| The determinant in this formula makes sense because (as noted above) when the set T has cardinality equal to the dimension n of the ambient space, the zonotope is a parallelotope. ==References==