Bipolar theorem

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In mathematics, the bipolar theorem is a theorem in convex analysis which provides necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.[1]:76–77

Statement of theorem[edit]

For any nonempty set in some linear space , then the bipolar set is given by

where denotes the convex hull.[1]:54[2]

Special case[edit]

is a nonempty closed convex cone if and only if when , where denotes the positive dual cone.[2][3]

Or more generally, if is a nonempty convex cone then the bipolar cone is given by

Relation to the Fenchel–Moreau theorem[edit]

Let

be the indicator function for a cone . Then the convex conjugate,

is the support function for , and . Therefore, if and only if .[1]:54[3]

References[edit]

  1. ^ a b c Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701.
  2. ^ a b Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 9780521833783. Retrieved October 15, 2011.
  3. ^ a b Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866.