# 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