# 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 ${\displaystyle C\subset X}$ in some linear space ${\displaystyle X}$, then the bipolar set ${\displaystyle C^{\circ \circ }=(C^{\circ })^{\circ }}$ is given by

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

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

### Special case

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

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

${\displaystyle C^{\circ \circ }=\operatorname {cl} C.}$

## Relation to the Fenchel–Moreau theorem

Let

${\displaystyle f(x)=\delta (x|C)={\begin{cases}0&x\in C\\\infty &{\text{else}}\end{cases}}}$

be the indicator function for a cone ${\displaystyle C}$. Then the convex conjugate,

${\displaystyle f^{*}(x^{*})=\delta (x^{*}|C^{\circ })=\delta ^{*}(x^{*}|C)=\sup _{x\in C}\langle x^{*},x\rangle }$

is the support function for ${\displaystyle C}$, and ${\displaystyle f^{**}(x)=\delta (x|C^{\circ \circ })}$. Therefore, ${\displaystyle C=C^{\circ \circ }}$ if and only if ${\displaystyle f=f^{**}}$.[1]:54[3]

## References

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.