proper cone

A proper cone is a cone (http://planetmath.org/Cone3) $C\subset {ℝ}^n$ that satisfies the following:

• $C$ is convex;

• $C$ is closed;

• $C$ is solid, meaning it has nonempty interior;

• $C$ is pointed, meaning $x,-x\in C\Rightarrow x=0$.

A proper cone $C$ induces a partial ordering on ${ℝ}^n$:

$$a⪯b\iff b-a\in C.$$

This ordering has many nice properties, such as transitivity, reflexivity, and antisymmetry.