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.