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→Dual cone: Changed to a standard (and consistent) definition of self-dual (cited); added examples of these.. |
Moved alternative definition of dual to follow existing one. Clarified that the conditions from Boyd & Vandenberghe assume this alternative definition. |
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<math>C^* </math> is always a [[convex cone]], even if <math>C </math> is neither [[convex set|convex]] nor a [[linear cone|cone]].
Alternatively, many authors define the dual cone in the context of a real Hilbert space, (such as <math>\mathbb{R}^n</math> equipped with the Euclidean inner product) to be what is sometimes called the <i>internal dual cone</i>.
When <math>C </math> is a cone, the following properties hold:<ref name="Boyd">{{cite book|title=Convex Optimization|first1=Stephen P.|last1=Boyd|first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83378-3|url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf|format=pdf|accessdate=October 15, 2011|pages=51-53}}</ref>▼
:<math>C^*_{internal} := \left \{y\in X: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \}
Using this latter definition for <math>C^*</math>, we have that
▲
* A non-zero vector <math>y</math> is in <math>C^*</math> if and only if both of the following conditions hold: (i) <math> y </math> is a [[surface normal|normal]] at the origin of a [[hyperplane]] that [[supporting hyperplane|supports]] <math>C </math>. (ii) <math> y </math> and <math>C </math> lie on the same side of that supporting hyperplane.
*<math>C^* </math> is [[closed set|closed]] and convex.
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A cone <math>C</math> in a vector space <math>X</math> is said to be ''self-dual'' if
<math>X</math> can be equipped with an [[inner product]]
<math>\langle . , . \rangle</math> such that the
internal dual cone is equal to <math>C</math><ref>Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.</ref>.
▲:<math>C^*_{internal} := \left \{y\in X: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \},</math>
▲is equal to <math>C</math><ref>Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.</ref>. Many authors <i>define</i> the dual cone in the context of a real Hilbert space, (such as <math>\mathbb{R}^n</math> equipped with the Euclidean inner product) to be what we have called the internal dual cone, and say a cone is self-dual if it is equal to its internal dual. (This is slightly different than the above definition, which permits a change of inner product. For instance, the above definition makes a cone in <math>\mathbb{R}^n</math> with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a with spherical base
▲in <math>\mathbb{R}^n</math> is equal to its internal dual.)
The nonnegative [[orthant]] of <math>\mathbb{R}^n</math> and the space of all [[positive semidefinite matrix|positive semidefinite matrices]] are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones"). So are all cones in <math>\mathbb{R}^3</math> whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in <math>\mathbb{R}^3</math> whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.
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