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A zonotope is a polytope which can be obtained as the Minkowski sum of finitely many closed line segments in
 . 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  -cube is the Minkowski sum of the line segments from the origin to the standard unit vectors  for
 . A hexagon is also a zonotope; for example, the Minkowski sum of the line segments based at the origin with endpoints at  ,  , and  is a hexagon. In fact, any projection of an  -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  -simplex is only a zonotope for  .
- 1
- Billera, L., R. Ehrenborg, and M. Readdy, The
 -index of zonotopes and arrangements, in Mathematical essays in honor of Gian-Carlo Rota, (B. E. Sagan and R. P. Stanley, eds.), Birkhäuser, Boston, 1998, pp. 23-40.
- 2
- Ziegler, G., Lectures on polytopes, Springer-Verlag, 1997.
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"zonotope" is owned by mps.
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Cross-references: pyramid, prism, projection, endpoints, hexagon, unit vectors, origin, line segments, Minkowski sum, unit, convex sets, compact, symmetric, hyperplane arrangements, finite, closed line segments, polytope
There are 2 references to this entry.
This is version 4 of zonotope, born on 2006-03-20, modified 2006-11-04.
Object id is 7749, canonical name is Zonotope.
Accessed 2184 times total.
Classification:
| AMS MSC: | 52B99 (Convex and discrete geometry :: Polytopes and polyhedra :: Miscellaneous) |
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Pending Errata and Addenda
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