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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.:76–77
Statement of theorem
Or more generally, if is a nonempty convex cone then the bipolar cone is given by
Relation to the Fenchel–Moreau theorem
- Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701.
- Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 9780521833783. Retrieved October 15, 2011.
- Rockafellar, R. Tyrrell (1997) . Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866.