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
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