# Convex cones

(Difference between revisions)
 Revision as of 20:49, 28 August 2008 (edit)← Previous diff Current revision (13:33, 24 November 2011) (edit) (undo) (14 intermediate revisions not shown.) Line 1: Line 1: - ==Nonorthogonal projection on extreme directons of convex cone== + We call the set $\mathcal{K}_{\!}\subseteq_{\!}\mathbb{R}^M$ a ''convex cone'' iff - ===pseudo coordinates=== + - Let $\mathcal{K}$ be a full-dimensional closed pointed convex cone + - in finite-dimensional Euclidean space $\mathbb{R}^n$. + - For any vector $\,v\,$ and a point $\,x\!\in\!\mathcal{K}$, + $\Gamma_{1\,},\Gamma_2\in\mathcal{K}~\Rightarrow~\zeta_{\,}\Gamma_1+_{_{}}\xi_{\,}\Gamma_2 - define [itex]\,d_v(x)\,$ to be the largest number $\,t^\star$ such that $\,x-t^{}v\!\in\!\mathcal{K}\,$. + \in_{_{}}\overline{\mathcal{K}}\textrm{~~for all~\,}\zeta_{\,},\xi\geq0.[/itex] - Suppose $\,x\,$ and $\,y\,$ are points in $\,\mathcal{K}\,$. + Apparent from this definition, $\zeta_{\,}\Gamma_{1\!}\in\overline{\mathcal{K}}$ + and $\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}$ for all $\zeta_{\,},\xi_{\!}\geq_{\!}0_{}$. - Further, suppose that $\,d_v(x)\!=_{\!}d_v(y)\,$ for every extreme direction $\,v\,$ of $\,\mathcal{K}\,$. + The set $\mathcal{K}$ is convex since, for any particular $\zeta_{\,},\xi\geq0$, - Then $\,x\,$ must be equal to $\,y\,$. + $\mu\,\zeta_{\,}\Gamma_1\,+\,(1-\mu)_{\,}\xi_{\,}\Gamma_2\in_{}\overline{\mathcal{K}}\quad\forall\,\mu\in_{}[0_{},1]$ - ===proof=== + because $\mu\,\zeta_{\,},(1-\mu)_{\,}\xi\geq0_{}$. + + 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''), + + $\mathcal{K}_\ell=\left\{\left[\begin{array}{c}x\\t\end{array}\right]\!\in\mathbb{R}^n\!\times\mathbb{R} + ~|~\|x\|_\ell\leq_{}t\right\}~,\qquad\ell\!=\!2$ + + 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 $\mathbb{R}_+$ in subspace $\mathbb{R}\,$, + any halfspace partially bounded by a hyperplane through the origin, + the positive semidefinite cone $\mathbb{S}_+^M$, + the cone of Euclidean distance matrices $\mathbb{EDM}^N$, + any subspace, and Euclidean vector space $\mathbb{R}^n$.

## 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 $LaTeX: \mathbb{R}^n$.