zonotope

A zonotope is a polytope which can be obtained as the Minkowski sum (http://planetmath.org/MinkowskiSum3) of finitely many closed line segments in $\mathbb{R}^{n}$ . Three-dimensional zonotopes are also sometimes called zonohedra. Zonotopes are dual to finite hyperplane arrangements. They are centrally symmetric, compact, convex sets.

For example, the unit $n$ -cube is the Minkowski sum of the line segments from the origin to the standard unit vectors $e_{i}$ for $1\leq i\leq n$ . A hexagon is also a zonotope; for example, the Minkowski sum of the line segments based at the origin with endpoints at $(1,0)$ , $(1,1)$ , and $(0,1)$ is a hexagon. In fact, any projection of an $n$ -cube is a zonotope.

The prism of a zonotope is always a zonotope, but the pyramid of a zonotope need not be. In particular, the $n$ -simplex (http://planetmath.org/HomologyTopologicalSpace) is only a zonotope for $n\leq 1$ .

References

1 Billera, L., R. Ehrenborg, and M. Readdy, The $\mathbf{cd}$ -index of zonotopes and arrangements, in Mathematical essays in honor of Gian-Carlo Rota, (B. E. Sagan and R. P. Stanley, eds.), BirkhÃÂ¤user, Boston, 1998, pp. 23–40.
2 Ziegler, G., Lectures on polytopes, Springer-Verlag, 1997.

Title	zonotope
Canonical name	Zonotope
Date of creation	2013-03-22 15:47:20
Last modified on	2013-03-22 15:47:20
Owner	mps (409)
Last modified by	mps (409)
Numerical id	7
Author	mps (409)
Entry type	Definition
Classification	msc 52B99
Synonym	zonohedron
Synonym	zonohedra
Related topic	HyperplaneArrangement