Combinatorial Properties of Associated Zonotopes

Let S 1 . . . ,Sr be r line segments, each of non-zero length, in n-dimensional euclidean space R n . If a polytope Z is defined as the vector (Minkowski) sum (1) Z = S 1 + . . . + Sr , then the segments Si will be called the components of Z. Since we do not wish to exclude the possibility that some of the components may be parallel, the polytope Z may be written in the form (1) in many different ways. For this reason it is convenient to define a zonotope to be the polytope Z together with some specified set of components {S1 , . . . , Sr }. Figures 1, 2 and 3 show some zonotopes of 1, 2 and 3 dimensions with 4, 5 and 6 components.