Convex cones

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<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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\in_{_{}}\overline{\mathcal{K}}\textrm{\;\;for all\;\,}\zeta_{\,},\xi\geq0.</math>
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\in_{_{}}\overline{\mathcal{K}}\textrm{\;\;for all\;\,}\zeta_{\,},\,\xi\geq0.</math>
Apparent from this definition, <math>\zeta_{\,}\Gamma_{1\!}\in\overline{\mathcal{K}}</math>
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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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>,
The set <math>\mathcal{K}</math> is convex since, for any particular <math>\zeta_{\,},\,\xi\geq0</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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