Polar set

See also polar set (potential theory).

In functional and convex analysis, related disciplines of mathematics, the polar set $A^{\circ }$ is a special convex set associated to any subset $A$ of a vector space $X$ lying in the dual space $X^{*}$. The bipolar of a subset is the polar of $A^{\circ }$, but lies in $X$ (not $X^{**}$).

Definitions

There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis. In each case, the definition describes a duality between certain subsets of a dual pair of (topological) vector spaces $(X,Y)$.

Geometric definition

Main article: Pole and polar

The polar cone of a convex cone $A\subseteq X$ is the set

$A^{\circ }:=\{y\in Y:\sup _{x\in A}|\langle x,y\rangle |\leq 0\}$

This definition gives a duality on points and hyperplanes, writing the latter as the intersection of two oppositely-oriented half-spaces. The polar hyperplane of a point $x\in X$ is the locus $\{y:\langle y,x\rangle =0\}$; the dual relationship for a hyperplane yields that hyperplane's polar point.

Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.

Functional analytic-definition

The polar of a set $A\subseteq X$ is the set

$A^{\circ }:=\{y\in Y:\sup _{x\in A}\langle x,y\rangle \leq 1\}$

This is an affine shift of the geometric definition; it has the useful characterization that the functional-analytic polar of the unit ball (in $X$) is precisely the unit ball (in $Y$).

Some authors include absolute values around the inner product; the two definitions coincide for circled sets.

Properties

• If $A\subseteq B$ then $B^{\circ }\subseteq A^{\circ }$
  • An immediate corollary is that $\bigcup _{i\in I}A_{i}^{\circ }\subseteq (\bigcap _{i\in I}A_{i})^{\circ }$; equality necessarily holds only for finitely-many terms.
• For all $\gamma \neq 0$ : $(\gamma A)^{\circ }={\frac {1}{\mid \gamma \mid }}A^{\circ }$.
• $(\bigcup _{i\in I}A_{i})^{\circ }=\bigcap _{i\in I}A_{i}^{\circ }$.
• For a dual pair $(X,Y)$ $A^{\circ }$ is closed in $Y$ under the weak-*-topology on $Y$.
• The bipolar $A^{\circ \circ }$ of a set $A$ is the closed convex hull of $A\cup \{0\}$, that is the smallest closed and convex set containing both $A$and $0$.
  • Similarly, the bidual cone of a cone $A$is the closed conic hull of $A$.
• For a closed convex cone $C$ in $X$, the dual cone is the polar of $C$; that is,

$C^{\circ }=\{y\in Y:\sup\{\langle x,y\rangle :x\in C\}\leq 0\}$

See also

• Polar cone
• Bipolar theorem