2. DEFINITIONS OF POLYHEDRA, THEIR FACES AND CONES 7
2.2. Faces of Polyhedron
Definition 2.7 (Face). Let V be an affine subspace in
Rn.
Given a set Π of
hyperplane in V , let P = P(Π) be a polyhedron in V . A subset F P is a face if
there exists a hyperplane πq,r in V (which does not have to be in Π) such that
F = πq,r P and P \ F
πq,r.+
(2.2)
We may replace P \ F by P, or πq,r + by (πq,r)◦ + in (2.2). Thus F is a face of P if and
only if there exists a vector q Rn and r R satisfying
q, u = r q, y for all u F and y P \ F. (2.3)
When F is a face of P, it is denoted by F P. The above hyperplane πq,r is called
the supporting hyperplane of the face F. The dimension of a face F of P is the
dimension of an ambient affine space Vam(F) of F where Vam(F) is defined in (2.1).
We denote the set of all k-dimensional faces of P by
Fk(P),
and
Fk(P)
by F(P).
By convention, an empty set is −1 dimensional face. Let dim(P) = m. Then we
call, a face F whose dimension is less than m, a proper face of P and denote it by
F P.
Definition 2.8 (Facet). Let P = P(Π) be a polyhedron in an affine space V
with dim(P) = dim(Vam(P)) = m. Then m 1 dimensional face F of P is called a
facet of P.
Example 2.3. Let P be the polyhedron defined in Example 2.1. Then for each
fixed k = 4, 5,
P = πqk,rk P =
j=4,5
(
πqj
,rj
P
)
is a 2 dimensional face of P. For each fixed k = 1, 2, 3,
Fk =πqk,rk P=
j=k,4
(πqj
,rj
P)=
j=k,5
(πqj
,rj
P)=
j=k,4,5
(πqj
,rj
P)=Ch(e , em)
with , m = k is a 1 dimensional face (edge) of P. For each fixed k, {1, 2, 3} with
k = ,
Fk, =
j=k,,
(πqj
,rj
P) =
j=k,,4
(πqj
,rj
P)
=
j=k,,5
(πqj
,rj
P) =
j=k,,4,5
(πqj
,rj
P) = {em}
with m = k, is a 0 dimensional face (vertex) of P.
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