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[[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<sup>o</sup>''. The dual cone and the polar cone are symmetric to each other with respect to the origin.]]
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==Dual cone==
The '''dual cone''' ''C{{sup|*}}'' of a [[subset]] ''C'' in a [[linear space]] ''X'', e.g. [[Euclidean space]] '''R'''<sup>''n''</sup>, 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>
where <''y'', ''x''> is the duality pairing between ''X'' and ''X{{sup|*}}'', i.e. <''y'', ''x''> = ''y''(''x'').
''C{{sup|*}}'' is always a [[convex cone]], even if ''C'' 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 '''R'''<sup>''n''</sup> equipped with the Euclidean inner product) to be what is sometimes called the ''internal dual cone''.
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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>
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=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf |format=pdf|accessdate=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:
#''y'' is a [[surface normal|normal]] at the origin of a [[hyperplane]] that [[supporting hyperplane|supports]] ''C''.
#''y'' and ''C'' lie on the same side of that supporting hyperplane.
*''C{{sup|*}}'' is [[closed set|closed]] and convex.
*''C''<sub>1</sub> ⊆ ''C''<sub>2</sub> implies <math>C_2^* \subseteq C_1^*</math>.
*If ''C'' has nonempty interior, then ''C{{sup|*}}'' is ''pointed'', i.e. ''C*'' contains no line in its entirety.
*If ''C'' is a cone and the closure of ''C'' is pointed, then ''C{{sup|*}}'' has nonempty interior.
*''C{{sup|**}}'' is the closure of the smallest convex cone containing ''C'' (a consequence of the [[hyperplane separation theorem]])
==Self-dual cones==
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:<math>C^o = \left \{y\in X^*: \langle y , x \rangle \leq 0 \quad \forall x\in C \right \}.</math>
It can be seen that the polar cone is equal to the negative of the dual cone, i.e. ''C<sup>o</sup>'' = −''C{{sup|*}}''.
For a closed convex cone ''C'' in ''X'', the polar cone is equivalent to the [[polar set]] for ''C''.<ref>{{cite book|last=Aliprantis |first=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>
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