Bipolar theorem explained

In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a 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]

Preliminaries

See main article: Polar set.

Suppose that

X is a topological vector space (TVS) with a continuous dual space X ^{\prime} and let \left\langle x, x ^{\prime} \right\rangle : = x ^{\prime} (x) for all x \in X and x ^{\prime} \in X ^{\prime} . The convex hull of a set A, denoted by \operatorname{co} A, is the smallest convex set containing A. The convex balanced hull of a set A is the smallest convex balanced set containing A.

The polar of a subset

A \subseteq X is defined to be: $$A^\backslash circ\; :=\; \backslash left\backslash .$$ while the prepolar of a subset B \subseteq X ^{\prime} is:$$^\; B\; :=\; \backslash left\backslash .$$The bipolar of a subset A \subseteq X, often denoted by A ^{\circ\circ} is the set $$A^\; :=\; ^\backslash left(A^\backslash right)\; =\; \backslash left\backslash .$$

Statement in functional analysis

Let

\sigma\left(X, X ^{\prime} \right) denote the weak topology on X (that is, the weakest TVS topology on A making all linear functionals in X ^{\prime} continuous).

The bipolar theorem: The bipolar of a subset

A \subseteq X is equal to the \sigma\left(X, X ^{\prime} \right) -closure of the convex balanced hull of A.

Statement in convex analysis

The bipolar theorem:^{[1] [2]} For any nonempty cone

A in some linear space X, the bipolar set A ^{\circ} is given by: $$A^\; =\; \backslash operatorname\; (\backslash operatorname\; \backslash ).$$

Special case

A subset

C \subseteq X is a nonempty closed convex cone if and only if C ^{++} = C ^{\circ} = C when C ^{++} = \left(C ^{+} \right) ^{+} , where A ^{+} denotes the positive dual cone of a set A. ^{[2] [3]} Or more generally, if C is a nonempty convex cone then the bipolar cone is given by $$C^\; =\; \backslash operatorname\; C.$$

Relation to the Fenchel–Moreau theorem

Let be the indicator function for a cone

C. Then the convex conjugate, $$f^*(x^*)\; =\; \backslash delta\backslash left(x^*|C^\backslash circ\backslash right)\; =\; \backslash delta^*\backslash left(x^*|C\backslash right)\; =\; \backslash sup\_\; \backslash langle\; x^*,x\; \backslash rangle$$ is the support function for C, and f ^{**} (x) = \delta(x|C ^{\circ\circ} ). Therefore, C = C ^{\circ} if and only if f = f ^{**} . ^{[1] [3]}

See also

• • − A generalization of the bipolar theorem.

Notes and References

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