Dual cone and polar cone: Difference between revisions

Revision as of 16:13, 11 May 2012

Revision as of 16:59, 4 June 2012

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:<math>C^* = \left \{y\in X^*: \langle y , x \rangle \geq 0 \quad \forall x\in C \right \},</math>

where <math>\langle y, x \rangle</math> is the duality pairing

between <math>X</math> and <math>X^*</math>, i.e. <math>\langle y, x \rangle = y(x) </math>.

<math>C^* </math> is always a [[convex cone]], even if <math>C </math> is neither [[convex set|convex]] nor a [[linear cone|cone]].

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*<math>C^{**} </math> is the closure of the smallest convex cone containing <math>C </math>.

==Self-dual cones==

A cone is said to be ''self-dual'' if <math>C = C^* </math>. The nonnegative [[orthant]] of <math>\mathbb{R}^n</math> and the space of all [[positive semidefinite matrix|positive semidefinite matrices]] are self-dual.

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 <i>internal dual cone</i> relative to this inner product,

:<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.

==Polar cone==