Convex cones

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We call the set <math>\mathcal{K}_{\!}\subseteq_{\!}\reals^M</math> a ''convex cone'' iff
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<math>\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2
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\in_{_{}}\overline{\mathcal{K}}\textrm{~~for all~\,}\zeta_{\,},\xi\geq0</math>
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Apparent from this definition, <math>\zeta_{\,}\Gamma_{1\!}\in\overline{\mathcal{K}}</math>
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and <math>\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}</math> for all <math>\zeta_{\,},\xi_{\!}\geq_{\!}0_{}</math>.
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The set <math>\mathcal{K}</math> is convex since, for any particular <math>\zeta_{\,},\xi\geq0</math>
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<math>\mu\,\zeta_{\,}\Gamma_1\,+\,(1-\mu)_{\,}\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}\quad\forall\,\mu\in_{}[0_{},1]</math>
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because <math>\mu\,\zeta_{\,},(1-\mu)_{\,}\xi\geq0_{}</math>.
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Obviously,
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<math>\{\mathcal{X}\}\supset\{\mathcal{K}\}</math>
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the set of all convex cones is a \emph{proper subset} of all cones.
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The set of convex cones is a narrower but more familiar class of cone, any member of which can be
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equivalently described as the intersection of
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a possibly (but not necessarily) infinite number of hyperplanes (through the origin)
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and halfspaces whose bounding hyperplanes pass through the origin; a halfspace-description.
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Interior of a convex cone is possibly empty.
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Familiar examples of convex cones include an unbounded ''ice-cream cone'' united with its interior
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(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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~|~\|x\|_\ell\leq_{}t\right\}~,\qquad\ell\!=\!2</math>
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and any polyhedral cone; ''e.g''., any orthant generated by Cartesian half-axes.
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Esoteric examples of convex cones include
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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 $\reals_+$ in subspace <math>\reals\,</math>,
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any halfspace partially bounded by a hyperplane through the origin,
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the positive semidefinite cone <math>\mathbb{S}_+^M</math>,
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the cone of Euclidean distance matrices,
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any subspace, and Euclidean vector space $\reals^n$.

Revision as of 18:33, 1 October 2008

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,

LaTeX: \{\mathcal{X}\}\supset\{\mathcal{K}\}

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),

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

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