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 $1\le i\le n$. 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 $n\le 1$.
References
- 1 Billera, L., R. Ehrenborg, and M. Readdy, The $\mathrm{cd}$-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 |