Convex cones

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m (Protected "Convex cones" [edit=autoconfirmed:move=autoconfirmed])
Current revision (12:33, 24 November 2011) (edit) (undo)
 
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We call the set <math>\mathcal{K}_{\!}\subseteq_{\!}\reals^M</math> a ''convex cone'' iff
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We call the set <math>\mathcal{K}_{\!}\subseteq_{\!}\mathbb{R}^M</math> a ''convex cone'' iff
<math>\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2
<math>\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2
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(a.k.a: ''second-order cone'', ''quadratic cone'', ''circular cone'', ''Lorentz cone''),
(a.k.a: ''second-order cone'', ''quadratic cone'', ''circular cone'', ''Lorentz cone''),
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<math>\mathcal{K}_\ell=\left\{\left[\begin{array}{c}x\\t\end{array}\right]\!\in\reals^n\!\times\reals
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<math>\mathcal{K}_\ell=\left\{\left[\begin{array}{c}x\\t\end{array}\right]\!\in\mathbb{R}^n\!\times\mathbb{R}
~|~\|x\|_\ell\leq_{}t\right\}~,\qquad\ell\!=\!2</math>
~|~\|x\|_\ell\leq_{}t\right\}~,\qquad\ell\!=\!2</math>
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Esoteric examples of convex cones include
Esoteric examples of convex cones include
the point at the origin, any line through the origin, any ray having the origin as base
the point at the origin, any line through the origin, any ray having the origin as base
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such as the nonnegative real line <math>\reals_+</math> in subspace <math>\reals\,</math>,
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such as the nonnegative real line <math>\mathbb{R}_+</math> in subspace <math>\mathbb{R}\,</math>,
any halfspace partially bounded by a hyperplane through the origin,
any halfspace partially bounded by a hyperplane through the origin,
the positive semidefinite cone <math>\mathbb{S}_+^M</math>,
the positive semidefinite cone <math>\mathbb{S}_+^M</math>,
the cone of Euclidean distance matrices <math>\mathbb{EDM}^N</math>,
the cone of Euclidean distance matrices <math>\mathbb{EDM}^N</math>,
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any subspace, and Euclidean vector space <math>\reals^n</math>.
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any subspace, and Euclidean vector space <math>\mathbb{R}^n</math>.

Current revision

We call the set LaTeX: \mathcal{K}_{\!}\subseteq_{\!}\mathbb{R}^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\mathbb{R}^n\!\times\mathbb{R}
~|~\|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: \mathbb{R}_+ in subspace LaTeX: \mathbb{R}\,, 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: \mathbb{R}^n.

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