Proper Real Cones

Definition

A cone $K \subset \R ^n$ is proper if it is (a) convex, (b) closed, (c) solid, and (d) pointed. A cone is solid if its interior is nonempty. A cone is pointed if it contains no line.

\[ x \in K, -x \in K \Rightarrow x = 0. \]

Example

The nonnegative orthant is a proper cone.