CHAPTER 2
Definitions of Polyhedra, Their Faces and Cones
In this chapter, we define basic notions of polyhedra, faces and their dual faces.
2.1. Polyhedron
Definition 2.1. Let U Rn be a subspace endowed with an inner product
, in Rn. Then V is called an affine subspace in Rn if V = p + U for some p Rn.
Definition 2.2. Let V be an affine subspace in
Rn.
A hyperplane in V is a
set
πq,r = {y V : q, y = r} where q
Rn
and r R.
The corresponding closed upper half-space and lower half-space are
πq,r
+
= {y V : q, y r} and πq,r

= {y V : q, y r}.
The open upper half-space and lower half space are
(πq,r)◦ +
= {y V : q, y r} and
(πq,r)◦
= {y V : q, y r}.
Definition 2.3 (Polyhedron in V ). Let V be an affine subspace in
Rn
and let
Π = {πqj
,rj
}j=1 N be a set of hyperplanes in V . A polyhedron P in V is defined to be
an intersection of closed upper half-spaces πqj +
,rj
:
P =
N
j=1
πqj
+
,rj
=
N
j=1
{y V : qj , y rj for 1 j N}.
We call the above collection Π = Π(P) the generator of P. We denote the polyhedron
P by P(Π) indicating its generator Π. Sometimes, we also mean the generator Π of
P to be the collection of normal vectors {qj}j=1 N instead of hyperplanes {πqj
,rj
}j=1.N
Example 2.1. A set of hyper-planes Π = {πqj
,rj
: 1 j 5} where qj =
1
2
(e1+
e2 + e3 −ej)
1
3
(e1 + e2 + e3) with rj = 1/6 for j = 1, 2, 3 and qj = ±(e1 + e2 + e3)
with rj = ±1 for j = 4, 5 makes a triangle P(Π) = Ch(e1, e2, e3) (two dimensional
polyhedron) in R3.
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