Bipolar theorem

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

For any nonempty set $C\subset X$ in some linear space $X$, then the bipolar cone $C^{{oo}}=(C^{o})^{o}$ is given by

$C^{{oo}}=\operatorname {cl}(\operatorname {co}\{\lambda c:\lambda \geq 0,c\in C\})$

where $\operatorname {co}$ denotes the convex hull.^[1]:54[2]

Special case

$C\subset X$ is a nonempty closed convex cone if and only if $C^{{++}}=C^{{oo}}=C$ when $C^{{++}}=(C^{+})^{+}$, where $(\cdot )^{+}$ denotes the positive dual cone.^[2][3]

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

$C^{{oo}}=\operatorname {cl}C.$

Relation to Fenchel–Moreau theorem

If $f(x)=\delta (x|C)={\begin{cases}0&{\text{if }}x\in C\\+\infty &{\text{else}}\end{cases}}$ is the indicator function for a cone $C$. Then the convex conjugate $f^{*}(x^{*})=\delta (x^{*}|C^{o})=\delta ^{*}(x^{*}|C)=\sup _{{x\in C}}\langle x^{*},x\rangle$ is the support function for $C$, and $f^{{**}}(x)=\delta (x|C^{{oo}})$. Therefore, $C=C^{{oo}}$ if and only if $f=f^{{**}}$.^[1]:54[3]

References

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