bipolar theorem

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.{{cite book |last1=Borwein |first1=Jonathan |authorlink1=Jonathan Borwein|last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=9780387295701}}{{rp|76–77}}

Preliminaries

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:

$A^\circ := \left\{ x^{\prime} \in X^{\prime} : \sup_{a \in A} \left| \left\langle a, x^{\prime} \right\rangle \right| \leq 1 \right\}.$

while the prepolar of a subset $B \subseteq X^{\prime}$ is:

${}^{\circ} B := \left\{ x \in X : \sup_{x^{\prime} \in B} \left| \left\langle x, x^{\prime} \right\rangle \right| \leq 1 \right\}.$

The bipolar of a subset $A \subseteq X,$ often denoted by $A^{\circ\circ}$ is the set

$A^{\circ\circ} := {}^{\circ}\left(A^{\circ}\right)$

= \left\{ x \in X : \sup_{x^{\prime} \in A^{\circ}} \left|\left\langle x, x^{\prime} \right\rangle\right| \leq 1 \right\}.

Statement in functional analysis

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

:The bipolar theorem:{{sfn|Narici|Beckenstein| 2011|pp=225-273}} 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:{{rp|54}}{{cite book|title=Convex Optimization|first1=Stephen P.|last1=Boyd|first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=9780521833783|url=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=65|format=pdf|accessdate=October 15, 2011|pages=51–53}} For any nonempty cone $A$ in some linear space $X,$ the bipolar set $A^{\circ \circ}$ is given by:

$A^{\circ \circ} = \operatorname{cl} (\operatorname{co} \{ r a : r \geq 0, a \in A \}).$

=Special case=

A subset $C \subseteq X$ is a nonempty closed convex cone if and only if $C^{++} = C^{\circ \circ} = C$ when $C^{++} = \left(C^{+}\right)^{+},$ where $A^{+}$ denotes the positive dual cone of a set $A.$ {{cite book|author=Rockafellar, R. Tyrrell|author-link=Rockafellar, R. Tyrrell|title=Convex Analysis|publisher=Princeton University Press|location=Princeton, NJ|year=1997|origyear=1970|isbn=9780691015866|pages=121–125}}

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

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