Zonotope Construction via Shephard's Theorem
Zonotope Construction via Shephard's Theorem
Given , a set of generators for , the zonotope generated by is the convex hull of all vectors of the form a,; that is, is the Minkowski sum of all segments , where . Also, is the shadow of the -dimensional cube via the projection .
X={,…,}
a
1
a
r
n
(X)
X
∑
a∈X
(X)
[0,a]
a∈X
(X)
r
r
[0,1]
(,…,)↦
t
1
t
r
r
∑
i=1
t
i
a
i
Given any such that is a basis of and , the Minkowski sum (S)=λ+(S) is called a parallelepiped. The zonotope is said to be paved by a collection of parallelepipeds whenever and is at most -dimensional for .
S⊂X
S
n
λ∈
n
λ
(X)
Π
(X)=
⋃
∈Π
⋂
(n-1)
≠∈Π
Shephard's theorem explicitly provides a recursively generated collection of vectors such that is a paving of . Each tile of this paving (i.e. each parallelepiped of the form ()=+()) is shown in the figure in a different color.
Λ={}
λ
i
()⊂X,abasisof
λ
i
S
i
S
i
S
i
n
(X)
λ
i
S
i
λ
i
S
i