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.

Statement of theorem

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

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

where $\operatorname {co}$ denotes the convex hull.

Special case

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

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

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

Relation to the Fenchel–Moreau theorem

Let

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

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

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

is the support function for $C$, and $f^{**}(x)=\delta (x|C^{\circ \circ })$. Therefore, $C=C^{\circ \circ }$ if and only if $f=f^{**}$.