User:MWinter4/Zonotopes

In mathematics, specifically in polytope theory, a zonotope is a convex polytope that saitsfies any of the following equivalent definitions:


 * it is the projection of a cube $$C_d:=[-1,1]^d$$.
 * it is the Minkowski sum of finitely many line segments.
 * all its faces are centrally symmetric.
 * all its 2-dimensional faces are centrally symmetric.
 * all its codmension $$\delta$$ faces are centrally symmetric for some fixes $$\delta\in\{1,...,d-2\}$$.
 * its normal fan is a central hyperplane arrangement.

All zonotopes are centally symmetric.

Special names for low-dimensional zonotopes are in use. A 2-dimensional zonotope is also called a zonogon, and a 3-dimensional zonotope is also called a zonohedron.

For a family of vectors $$v_1,...,v_n$$, the generated zonotope is given by


 * $$Z=\mathrm{Zon}(v_1,...,v_n):=\sum_{i=1}^n [-v_i,v_i] = \{\alpha_1 v_1+\cdots + \alpha_n v_n\mid \alpha\in[-1,1]^n\}.$$

Here the $$[v_i,-v_i]:=\mathrm{conv}(\{v_i,-v_i\})$$ represent the line segments whose Minkowski sum makes up the zonotope.

Zonotopes form a hereditary family of polytopes, as all faces of a zonotope are again zonotopes. The dual of a zonotope of dimension $$d\ge 3$$ is never a zonotope.

The Hausdorff limit of a sequence of zonotopes is a zonoid. It can be shown that the construction of projection bodies defines a bijection between the class of general centrally symmetric convex bodies and zonoids.

Examples
Zonotopes are very rich in structure as the Minkowski sum of any generic family of line segments defines a zonotope. There are however some especially well known zonotopes.


 * Every centrally symmetric polygon is a zonotope.
 * The $$d$$-cube $$[-1,1]^d$$ is a zonotope.
 * Every prism with a centrally symmetric base is a zonotope.
 * More generally, the Minkowski sum of two zonotopes is a zonotope.
 * The permutahedron is a zonotope in every dimension. In dimension three this is the truncated octahedron. More generally, W-permutahedra (where W is a reflection group) are zonotopes. These are also known as omnitruncated uniform polytopes.

Non-examples
Any polytope that is not centrally symmetric or has a face that is not centrally symmetric, such as any Platonic solid other than the cube. Note that a polytope is necessarily already a zonotope if all its $$\delta$$-dimensional faces are centrally symmetric for $$\delta\in\{2,...,d-2\}$$. The 24-cell is a polytope all whose facets are centrally symmetric, yet it is not a zonotope.

Faces of zonotopes
Given a finite set $$V:=\{v_1,...,v_n\}\subset\Bbb R^d$$ of vectors. Let $$Z:=\mathrm{Zon}(V)$$ be the generated zonotope.

The faces of $$Z$$ are in one-to-one correspondene with the so-called flats of $$V$$.

Properties

 * Maximal number of faces of generic zonotope etc.

Many open problems are either simple or have been solved for zonotopes.


 * The combinatorics of a zonotope can be reconstructed from tis edge graph.
 * A zonotope (other than a parallelepiped) can be covered by (4/3)^n smaller copies of itself.
 * Kalai's 3^d conjecture is trivial for zonotopes.
 * Mahler's conjecture is proven for zonotopes.

Zonotopes, hyperplane arrangements and matroids
Zonotopes relate to general polytopes as matroids relate to oriented matroids.

Fix a zonotope $$Z$$ defined from the set of vectors $$V = \{v_1,\dots,v_n\}\subset\mathbb{R}^d$$ and let $$M$$ be the $$d \times n$$ matrix whose columns are the $$v_i$$. Then the vector matroid $$\underline{\mathcal{M}}$$ on the columns of $$M$$ encodes a wealth of information about $$Z$$, that is, many properties of $$Z$$ are purely combinatorial in nature.

For example, pairs of opposite facets of $$Z$$ are naturally indexed by the cocircuits of $$\mathcal{M}$$ and if we consider the oriented matroid $$\mathcal{M}$$ represented by $${M}$$, then we obtain a bijection between facets of $$Z$$ and signed cocircuits of $$\mathcal{M}$$ which extends to a poset anti-isomorphism between the face lattice of $$Z$$ and the covectors of $$\mathcal{M}$$ ordered by component-wise extension of $$0 \prec +, -$$. In particular, if $$M$$ and $$N$$ are two matrices that differ by a projective transformation then their respective zonotopes are combinatorially equivalent. The converse of the previous statement does not hold: the segment $$[0,2] \subset \mathbb{R}$$ is a zonotope and is generated by both $$\{2\mathbf{e}_1\}$$ and by $$\{\mathbf{e}_1, \mathbf{e}_1\}$$ whose corresponding matrices, $$[2]$$ and $$[1~1]$$, do not differ by a projective transformation.

Zonohedra
Zonohedron is the specific name for a 3-dimensional zonotope. It derives from the word "polyhedron", which some authors use specifically for 3-dimensional polytopes.

Inscribed zonotopes
It is easy to construct inscribed zonotopes in dimension two. Every regular $$2n$$-gon is an inscribed zonotope. In higher dimension this is much more restrictive. Besides cubes and prisms over 2-dimensional inscribed zonotopes, further examples are provided by some uniform polytopes. More precisely, one can choose use certain generic orbit polytopes of reflection groups.

excluding prisms, there seem to exist exactly 17 inscribed zonotopes in dimension three. These are constructed as projections of higher-dimensional uniform inscribed zonotopes along faces. They also correspond to certain simplicial hyperplane arrangement, which are also very rare. However, not every simplicial hyperplane arrangement gives rise to an inscribed polytope.