Dual cone and polar cone: Difference between revisions

Revision as of 07:12, 16 November 2014

Revision as of 17:00, 27 March 2015

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where <''y'', ''x''> is the duality pairing between ''X'' and ''X*'', i.e. <''y'', ''x''> = ''y''(''x'').

''C*'' 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'''^''n'' equipped with the Euclidean inner product) to be what is sometimes called the ''internal dual cone''.

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==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>