Dual cone and polar cone: Difference between revisions

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'''Dual cone''' and '''polar cone''' are closely related concepts in [[convex analysis]], a branch of [[mathematics]].

== Dual cone ==

The '''dual cone''' ''C{{sup|*}}'' of a [[subset]] ''C'' in a [[linear space]] ''X'', e.g. [[Euclidean space]] '''R'''^''n'', with [[topological]] [[dual space]] ''X{{sup|*}}'' is the set

=== In a vector space ===

The '''dual cone''' ''C{{sup|*}}'' of a [[subset]] ''C'' in a [[linear space]] ''X'' over the [[real numbers|real]]s, e.g. [[Euclidean space]] '''R'''^''n'', with [[topological]] [[dual space]] ''X{{sup|*}}'' is the set

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

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''C{{sup|*}}'' is always a [[convex cone]], even if ''C'' is neither [[convex set|convex]] nor a [[linear cone|cone]].

=== In a topological vector space ===

If ''X'' is a [[topological vector space]] over the real or complex numbers, then the '''dual cone''' of a subset ''C'' ⊆ ''X'' is the following set of continuous linear functionals on ''X'':

:<math>C^{\prime} := \left\{ f \in X^{\prime} : \operatorname{Re} \left( f (x) \right) \geq 0 \text{ for all } x \in C \right\}</math>.{{sfn | Schaefer | 1999 | pp=215–222}}

No matter what ''C'' is, ''C{{sup|\prime}}'' will be a convex cone.

If ''C'' ⊆ {0} then <math>C^{\prime} = X^{\prime}</math>.

=== In a Hilbert space (internal dual cone) ===

Alternatively, many authors define the dual cone in the context of a real [[Hilbert space]] (such as '''R'''^''n'' equipped with the Euclidean inner product) to be what is sometimes called the ''internal dual cone''.

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*''C{{sup|**}}'' is the closure of the smallest convex cone containing ''C'' (a consequence of the [[hyperplane separation theorem]])

== Self-dual cones ==

A cone ''C'' in a vector space ''X'' is said to be ''self-dual'' if ''X'' can be equipped with an [[inner product]] ⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to ''C''.<ref>Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.</ref> Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that 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 '''R'''^''n'' with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in '''R'''^''n'' is equal to its internal dual.

A cone ''C'' in a vector space ''X'' is said to be ''self-dual'' if ''X'' can be equipped with an [[inner product]] ⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to ''C''.<ref>Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.</ref>

The nonnegative [[orthant]] of '''R'''^''n'' 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 '''R'''³ whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in '''R'''³ 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.

Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that 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 '''R'''^''n'' with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in '''R'''^''n'' is equal to its internal dual.

The nonnegative [[orthant]] of '''R'''^''n'' 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 '''R'''³ whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in '''R'''³ 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.

So are all cones in '''R'''³ whose base is the convex hull of a regular polygon with an odd number of vertices.

A less regular example is the cone in '''R'''³ 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 ==

==Polar cone==

[[File:Polar cone illustration.svg|right|thumb|The polar of the closed convex cone ''C'' is the closed convex cone ''C^o'', and vice versa.]]

For a set ''C'' in ''X'', the '''polar cone''' of ''C'' is the set<ref name="Rockafellar">{{cite book|author=[[Rockafellar, R. Tyrrell]]|title=Convex Analysis | publisher=Princeton University Press |___location=Princeton, NJ|year=1997|origyear=1970|isbn=978-0-691-01586-6|pages=121–122}}</ref>

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== See also ==

* [[Bipolar theorem]]

* [[Polar set]]

== References ==

{{Reflist}}

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| isbn = 0-8218-1990-9

}}

* {{cite book | last=Schaefer | first=Helmut H.| title=Topological Vector Spaces | publisher=Springer New York Imprint Springer | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | publication-place=New York, NY | year=1999 | isbn=978-1-4612-7155-0 | oclc=840278135 | ref=harv}} <!-- {{sfn | Schaefer | 1999 | p=}} -->

[[Category:Convex geometry]]