Dual cone and polar cone: Difference between revisions

Revision as of 23:07, 2 November 2022

Revision as of 13:44, 20 December 2023

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*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]])

== Properties ==

If C is a convex cone and C* is its dual cone, then:<ref name=":0">{{Cite book |last=Arkadi Nemirovsky |url=https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=8c3cb6395a35cb504019f87f447d65cb6cf1cdf0 |title=Interior point polynomial-time methods in convex programming |year=2004}}</ref>

* C* is a closed convex cone.

* C** equals C.

* C is [[Pointed cone|pointed]] iff C* has a nonempty interior, and C has a nonempty interior iff C* is pointed.

== Self-dual cones ==