# Convex cones

### From Wikimization

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+ | We call the set <math>\mathcal{K}_{\!}\subseteq_{\!}\reals^M</math> a ''convex cone'' iff | ||

+ | <math>\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2 | ||

+ | \in_{_{}}\overline{\mathcal{K}}\textrm{~~for all~\,}\zeta_{\,},\xi\geq0</math> | ||

+ | |||

+ | Apparent from this definition, <math>\zeta_{\,}\Gamma_{1\!}\in\overline{\mathcal{K}}</math> | ||

+ | and <math>\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}</math> for all <math>\zeta_{\,},\xi_{\!}\geq_{\!}0_{}</math>. | ||

+ | The set <math>\mathcal{K}</math> is convex since, for any particular <math>\zeta_{\,},\xi\geq0</math> | ||

+ | |||

+ | <math>\mu\,\zeta_{\,}\Gamma_1\,+\,(1-\mu)_{\,}\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}\quad\forall\,\mu\in_{}[0_{},1]</math> | ||

+ | |||

+ | because <math>\mu\,\zeta_{\,},(1-\mu)_{\,}\xi\geq0_{}</math>. | ||

+ | |||

+ | Obviously, | ||

+ | |||

+ | <math>\{\mathcal{X}\}\supset\{\mathcal{K}\}</math> | ||

+ | |||

+ | the set of all convex cones is a \emph{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. | ||

+ | Interior of a convex cone is possibly empty. | ||

+ | |||

+ | |||

+ | 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''), | ||

+ | |||

+ | <math>\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</math> | ||

+ | |||

+ | 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 $\reals_+$ in subspace <math>\reals\,</math>, | ||

+ | any halfspace partially bounded by a hyperplane through the origin, | ||

+ | the positive semidefinite cone <math>\mathbb{S}_+^M</math>, | ||

+ | the cone of Euclidean distance matrices, | ||

+ | any subspace, and Euclidean vector space $\reals^n$. |

## Revision as of 18:33, 1 October 2008

We call the set a *convex cone* iff

Apparent from this definition, and for all . The set is convex since, for any particular

because .

Obviously,

the set of all convex cones is a \emph{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. Interior of a convex cone is possibly empty.

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*),

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 $\reals_+$ in subspace ,
any halfspace partially bounded by a hyperplane through the origin,
the positive semidefinite cone ,
the cone of Euclidean distance matrices,
any subspace, and Euclidean vector space $\reals^n$.