6 JOONIL KIM
Definition 2.4. Let B = {q1, · · · , qM }
Rn.
Then the span of B is the set
Sp(B) =



M
j=1
cjqj : cj R



.
The convex span of B and its interior are defined by
CoSp(B) =



M
j=1
cjqj : cj 0



and
CoSp◦(B)
=



M
j=1
cjqj : cj 0



respectively. Finally the convex hull of B is the set
Ch(B) =



M
j=1
cjqj : cj 0 and
M
j=1
cj = 1



.
If B Rn is not a finite set, then the span of B is defined by the collection of all
finite linear combinations of vectors in B.
Definition 2.5 (Ambient Space of Polyhedron). Let P
Rn
and p, q P.
Then
Sp(P p) = Sp(P q) for all p, q P.
We denote the vector space Sp(P p) by V (P). The dimension of P is defined by
dim(P) = dim(V (P)).
From the fact p q V (P),
V (P) + p = V (P) + q.
We call V (P) + p the ambient affine space of P in Rn and denote it by Vam(P) :
Vam(P) = V (P) + p, (2.1)
which is the smallest affine space containing P.
Example 2.2. Let P be the polyhedron defined in Example 2.1. Then V (P) =
{y : e1 + e2 + e3, y = 0} and Vam = {y : e1 + e2 + e3, y = 1}.
Definition 2.6. Let B
Rn.
Then the rank of a set B is the number of
linearly independent vectors in B:
rank(B) = dim(Sp(B)).
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