Dual cone and polar cone: Difference between revisions

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{{Short description|conceptsConcepts in convex analysis}}

[[File:Dual cone illustration.svg|right|thumb|A set ''C'' and its dual cone ''C{{sup|*}}''.]]

[[File:Polar cone illustration1.svg|right|thumb|A set ''C'' and its polar cone ''C^o''. The dual cone and the polar cone are symmetric to each other with respect to the origin.]]

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where <math>\langle y, x \rangle</math> is the [[dual system|duality pairing]] between ''X'' and ''X{{sup|*}}'', i.e. <math>\langle y, x\rangle = y(x)</math>.

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

=== In a topological vector space ===

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

=== Properties ===

Using this latter definition for ''C{{sup|*}}'', we have that when ''C'' 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=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=65 |format=pdf|access-date=October 15, 2011|pages=51–53}}</ref>

* A non-zero vector ''y'' is in ''C{{sup|*}}'' if and only if both of the following conditions hold:

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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 thanfrom 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.

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It can be seen that the polar cone is equal to the negative of the dual cone, i.e. ''C^o'' = −''C{{sup|*}}''.

For a closed convex cone ''C'' in ''X'', the polar cone is equivalent to the [[polar set]] for ''C''.<ref>{{cite book|lastlast1=Aliprantis |firstfirst1=C.D.|last2=Border |first2=K.C. |title=Infinite Dimensional Analysis: A Hitchhiker's Guide|edition=3|publisher=Springer|year=2007|isbn=978-3-540-32696-0|doi=10.1007/3-540-29587-9|page=215}}</ref>

== See also ==

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| last = Ramm

| first = A.G.

|editor=Shivakumar, P.N. |editor2=Strauss, A.V.

| title = Operator theory and its applications

| publisher = Providence, R.I.: American Mathematical Society

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* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer|Wolff| 1999 | p=}} -->

{{Ordered topological vector spaces}}

{{OrderedTopologicalVectorSpaces}}

[[Category:Convex analysis]]

[[Category:Convex geometry]]

[[Category:Linear programming]]

[[Category:Convex analysis]]