Convex cones
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| - | + | We call the set <math>\mathcal{K}_{\!}\subseteq_{\!}\mathbb{R}^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 | ||
| + | \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> | ||
| + | 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\geq0\,,\;(1-\mu)_{\,}\xi\geq0_{}</math>. | ||
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| + | Obviously, the set of all convex cones is a 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 | ||
| + | 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. | ||
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| + | Convex cones need not be full-dimensional. | ||
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| + | 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''), | ||
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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> | ||
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| + | 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 <math>\mathbb{R}_+</math> in subspace <math>\mathbb{R}\,,\,</math> | ||
| + | any halfspace partially bounded by a hyperplane through the origin, | ||
| + | the positive semidefinite cone <math>\mathbb{S}_+^M,\,</math> | ||
| + | the cone of Euclidean distance matrices <math>\mathbb{EDM}^N,\,</math> | ||
| + | any subspace, and Euclidean vector space <math>\mathbb{R}^n</math>. | ||
Current revision
We call the set a convex cone iff
Apparent from this definition,
and
for all
.
The set is convex since, for any particular
,
because .
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),
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 in subspace
any halfspace partially bounded by a hyperplane through the origin,
the positive semidefinite cone
the cone of Euclidean distance matrices
any subspace, and Euclidean vector space
.