Real Cones

Definition

A set $C \subset \R ^n$ is a cone (or nonegative homogeneous)

\[ x \in C \text{ and } \theta \geq 0 \Rightarrow \theta x \in C. \]

Let $x \in \R ^n$. Define $C_1 \subset \R ^n$ by

\[ C_1 = \Set{\theta x}{\theta \geq \theta }. \]

$C_1$ is a cone. The set

\[ C_2 = \Set{x \in \R ^{n}}{x_i \geq 0 \text{ for } i = 1, \dots , n} \]

is a cone. $C_2$ is called the non-negative orthant. The set

\[ C_3 = \Set{x \in \R ^{n}}{x_i \leq 0 \text{ for } i = 1, \dots , n} \]

is a cone. The set $C_2 \cup C_3$ is a cone.

We denote the nonnegative orthant of $\R^n$ by $\R ^n_+$. We denote the nonpositive orthant of $\R^n$ by $\R ^n_-$.