8 JOONIL KIM

Definition 2.9. Let F be a face of a convex polyhedron P. Then the boundary

∂F of F is defined to be G, where the union is over all faces G F. When

dim(F) = k,

∂F =

dimG=k−1,G F

G,

since faces whose dimensions k − 1 are contained on k − 1 dimensional faces of

F. Note that ∂F is the boundary of F with respect to the usual topology of Vam(F)

in (2.1).

Example 2.4. Let a polyhedron P and its faces Fk be defined as in Examples

2.1 through 2.3. Then the boundary ∂P of P is given by ∂P =

dimF=1,F P

F =

F1 ∪ F2 ∪ F3 ⊂

π∈Π

π.

Definition 2.10. Let F be a face of a convex polyhedron P. Then the interior

F◦

of P is defined to be

F◦

= F \ ∂F. Note also that

F◦

is the interior of F with

respect to the usual topology defined on Vam(F) in (2.1).

Example 2.5. Observe that CoSp(p1, · · · , pN )◦ =

∑N

j=1

αjpj : αj 0 .

2.3. A Cone (Dual Face) of Face

Definition 2.11 (Dual face). Let F be a face of a polyhedron P in

Rn.

Then

the cone (dual face)

F∗

of F is defined by

F∗|P

= {q ∈

Rn

: ∃ r ∈ R such that F ⊂ πq,r ∩ P and P \ F ⊂

πq,r}+

= {q ∈

Rn

: ∃ r ∈ R such that q, u = r ≤ q, y for all u ∈ F, y ∈ P \ F}. (2.4)

The interior of a cone (dual face)

F∗

is the set of all nonzero normal vectors q

satisfying (2.2):

(F∗)◦|P

= {q ∈

Rn

: ∃ r ∈ R such that F = πq,r ∩ P and P \ F ⊂

πq,r}+

= {q ∈

Rn

: ∃ r ∈ R such that F = πq,r ∩ P and P \ F ⊂

(πq,r)◦}+

(2.5)

=

{q∈Rn

: ∃ r ∈R such that q, u =r q, y for all u∈F, y∈P \ F}.

We use the notation

F∗|(P,V

) when we restrict q in a given vector space V . Thus

F∗|P

=

F∗|(P, Rn)

in (2.4). If there is no ambiguity, we write just

F∗

instead of

F∗|P

or

F∗|(P, Rn).

We note that

F∗

itself is a polyhedron in

Rn

and

(F∗)◦

is an interior

of

F∗.

Remark 2.1. To understand a cone

F∗

as a dual face of F, one is likely to define

a cone of F by the collection of all normal vectors q satisfying (2.2) as in (2.5). If