Dual cone and polar cone: Difference between revisions

Revision as of 18:29, 18 June 2018

Revision as of 13:00, 7 December 2018

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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 ''X'' and ''X{{sup|*}}'', i.e. <''math>\langle y'', ''x''>\rangle = ''y''(''x'')</math>.

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

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#''y'' and ''C'' lie on the same side of that supporting hyperplane.

*''C{{sup|*}}'' is [[closed set|closed]] and convex.

*''C''<sub>1</submath>C_1 ⊆\subseteq ''C''<sub>2C_2</submath> 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.