# Convex cones

(Difference between revisions)
 Revision as of 01:06, 14 June 2010 (edit) (TaADEuuWqBB)← Previous diff Revision as of 03:11, 14 June 2010 (edit) (undo) (Undo revision 2060 by 217.79.148.78 (Talk))Next diff → Line 1: Line 1: - KiRecr rgvflmafyjjc, [url=http://jwkmjcnjmsre.com/]jwkmjcnjmsre[/url], [link=http://csnagzwcxzqi.com/]csnagzwcxzqi[/link], http://ixelspdisfvg.com/ + We call the set \mathcal{K}_{\!}\subseteq_{\!}\reals^M[/itex] a ''convex cone'' iff + + $\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2 + \in_{_{}}\overline{\mathcal{K}}\textrm{~~for all~\,}\zeta_{\,},\xi\geq0. + + Apparent from this definition, [itex]\zeta_{\,}\Gamma_{1\!}\in\overline{\mathcal{K}} + and [itex]\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}$ for all $\zeta_{\,},\xi_{\!}\geq_{\!}0_{}$. + + The set [itex]\mathcal{K} is convex since, for any particular [itex]\zeta_{\,},\xi\geq0, + + [itex]\mu\,\zeta_{\,}\Gamma_1\,+\,(1-\mu)_{\,}\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}\quad\forall\,\mu\in_{}[0_{},1] + + because [itex]\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''), + + [itex]\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 [itex]\reals_+ in subspace [itex]\reals\,, + any halfspace partially bounded by a hyperplane through the origin, + the positive semidefinite cone [itex]\mathbb{S}_+^M, + the cone of Euclidean distance matrices [itex]\mathbb{EDM}^N, + any subspace, and Euclidean vector space [itex]\reals^n.

## Revision as of 03:11, 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 $LaTeX: \reals^n$.