# 9.6. Cones III#

## 9.6.1. Polyhedral Cones#

Definition 9.37 (Polyhedral cone)

The conic hull of a finite set of points is known as a polyhedral cone.

In other words, let $$\{ \bx_1, \bx_2, \dots, \bx_m \}$$ be a finite set of points. Then

$C = \cone \{ \bx_1, \bx_2, \dots, \bx_m \}$

is known as a polyhedral cone.

Theorem 9.65

A polyhedral cone is nonempty, closed and convex.

Proof. Since it is the conic hull of a nonempty set, hence it is nonempty and convex.

Theorem 9.131 shows that the conic hulls of a finite set of points are closed.

Remark 9.7 (Polyhedral cone alternative formulations)

Following are some alternative definitions of a polyhedral cone.

1. A cone is polyhedral if it is the intersection of a finite number of half spaces which have $$\bzero$$ on their boundary.

2. A cone $$C$$ is polyhedral if there is some matrix $$\bA$$ such that $$C = \{ \bx \in \RR^n \ST \bA \bx \succeq \bzero \}$$.

3. A cone is polyhedral if it is the solution set of a system of homogeneous linear inequalities.

### 9.6.1.1. Polar Cones#

Theorem 9.66 (Polar cone of a polyhedral cone)

Let the ambient space by $$\RR^n$$. Let $$\bA \in \RR^{m \times n}$$. Let

$C = \{\bx \in \RR^n \ST \bA \bx \preceq \bzero \}.$

Then

$C^{\circ} = \{ \bA^T \bt \ST \bt \in \RR^m_+ \}.$

We note that the set $$C$$ is a convex cone. It is known as the convex polyhedral cone.

Proof. We note that $$\by \in C^{\circ}$$ if and only if $$\bx^T \by \leq 0$$ for every $$\bx$$ satisfying $$\bA \bx \preceq \bzero$$.

1. Thus, for every $$\bx \in \RR^n$$, the statement $$\bA \bx \preceq \bzero \implies \bx^T \by \leq 0$$ is true.

2. By Farkas’ lemma (Theorem 10.55), it is equivalent to the statement that there exists $$\bt \succeq \bzero$$ such that $$\bA^T \bt = \by$$.

3. Thus,

$C^{\circ} = \{ \bA^T \bt \ST \bt \in \RR^m_+ \}.$