proper cone

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

 $a\preceq b\Leftrightarrow b-a\in C.$

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

References

• 1 S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
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