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.