On zonotopes

Abstract

In this paper is described a diagram technique for zonotopes, or vector sums of line segments, which is analogous to that of Gale diagrams for general polytopes, and central diagrams for centrally symmetric polytopes. The use of these new zonal diagrams leads to relationships between zonotopes with n n zones of dimensions d d and n − d n - d , and enables one to enumerate all the combinatorial types of d d -zonotopes with n ≦ d + 2 n \leqq d + 2 zones. The connexion between arrangements of hyperplanes in projective space and zonotopes leads to corresponding new results about arrangements.