# Convex cones

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## Revision as of 02:11, 14 June 2010

We call the set $LaTeX: \mathcal{K}_{\!}\subseteq_{\!}\reals^M$ a convex cone iff

$LaTeX: \Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2 \in_{_{}}\overline{\mathcal{K}}\textrm{~~for all~\,}\zeta_{\,},\xi\geq0.$

Apparent from this definition, $LaTeX: \zeta_{\,}\Gamma_{1\!}\in\overline{\mathcal{K}}$ and $LaTeX: \xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}$ for all $LaTeX: \zeta_{\,},\xi_{\!}\geq_{\!}0_{}$.

The set $LaTeX: \mathcal{K}$ is convex since, for any particular $LaTeX: \zeta_{\,},\xi\geq0$,

$LaTeX: \mu\,\zeta_{\,}\Gamma_1\,+\,(1-\mu)_{\,}\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}\quad\forall\,\mu\in_{}[0_{},1]$

because $LaTeX: \mu\,\zeta_{\,},(1-\mu)_{\,}\xi\geq0_{}$.

Obviously, the set of all convex cones is a proper subset of all cones.

The set of convex cones is a narrower but more familiar class of cone, any member of which can be equivalently described as the intersection of a possibly (but not necessarily) infinite number of hyperplanes (through the origin) and halfspaces whose bounding hyperplanes pass through the origin; a halfspace-description.

Convex cones need not be full-dimensional.

Familiar examples of convex cones include an unbounded ice-cream cone united with its interior (a.k.a: second-order cone, quadratic cone, circular cone, Lorentz cone),

$LaTeX: \mathcal{K}_\ell=\left\{\left[\begin{array}{c}x\\t\end{array}\right]\!\in\reals^n\!\times\reals ~|~\|x\|_\ell\leq_{}t\right\}~,\qquad\ell\!=\!2$

and any polyhedral cone; e.g., any orthant generated by Cartesian half-axes. Esoteric examples of convex cones include the point at the origin, any line through the origin, any ray having the origin as base such as the nonnegative real line $LaTeX: \reals_+$ in subspace $LaTeX: \reals\,$, any halfspace partially bounded by a hyperplane through the origin, the positive semidefinite cone $LaTeX: \mathbb{S}_+^M$, the cone of Euclidean distance matrices $LaTeX: \mathbb{EDM}^N$, any subspace, and Euclidean vector space $LaTeX: \reals^n$.