A Brief on Zonotopes

A zonotope is the Minkowski sum of finitely many line segments:

*Z=l _{1}+l_{2}+...+l_{n }*

A "typical" zonotope, in this case a sum of 14 line segments, is displayed on the right.

There are many equivalent definitions. Another algebraic definition
closely related to ours is to say that *Z* is the linear image of
an hypercube in *n*-space. This is evident by writing the line segments
as *l _{k}=*[

*Z=C *[*-1,1*]^{n},

where *C* is a matrix with *n* column vectors *c _{k }*and
[

Every face of a zonotope is again a zonotope. The collection of all
faces which have a given line segment as a common summand form a band (or
zone; etymology) around the zonotope. For less than four line segments,
the zonotope *Z* is simply a line segment (*n=1*), a parallelogram
(*n=2*) or a parallelepiped (*n=3*). For more than three segments,
the geometry is more complex. Check out the Online Zonotope Builder and Viewer.