# Polar set

In functional analysis and related areas of mathematics the polar set of a given subset of a vector space is a certain set in the dual space.

Given a dual pair ${\displaystyle (X,Y)}$ the polar set or polar of a subset ${\displaystyle A}$ of ${\displaystyle X}$ is a set ${\displaystyle A^{\circ }}$ in ${\displaystyle Y}$ defined as

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

The bipolar of a subset ${\displaystyle A}$ of ${\displaystyle X}$ is the polar of ${\displaystyle A^{\circ }}$. It is denoted ${\displaystyle A^{\circ \circ }}$ and is a set in ${\displaystyle X}$.

## Properties

• ${\displaystyle A^{\circ }}$ is absolutely convex
• If ${\displaystyle A\subseteq B}$ then ${\displaystyle B^{\circ }\subseteq A^{\circ }}$
• So ${\displaystyle \bigcup _{i\in I}A_{i}^{\circ }\subseteq (\bigcap _{i\in I}A_{i})^{\circ }}$, where equality of sets does not necessarily hold.
• For all ${\displaystyle \gamma \neq 0}$ : ${\displaystyle (\gamma A)^{\circ }={\frac {1}{\mid \gamma \mid }}A^{\circ }}$
• ${\displaystyle (\bigcup _{i\in I}A_{i})^{\circ }=\bigcap _{i\in I}A_{i}^{\circ }}$
• For a dual pair ${\displaystyle (X,Y)}$ ${\displaystyle A^{\circ }}$ is closed in ${\displaystyle Y}$ under the weak-*-topology on ${\displaystyle Y}$
• The bipolar ${\displaystyle A^{\circ \circ }}$ of a set ${\displaystyle A}$ is the absolutely convex envelope of ${\displaystyle A}$, that is the smallest absolutely convex set containing ${\displaystyle A}$. If ${\displaystyle A}$ is already absolutely convex then ${\displaystyle A^{\circ \circ }=A}$.
• For a closed convex cone ${\displaystyle C}$ in ${\displaystyle X}$, the polar cone is equivalent to the one-sided polar set for ${\displaystyle C}$, given by
${\displaystyle C^{\circ }=\{y\in Y:\sup\{\langle x,y\rangle :x\in C\}\leq 1\}=\{y\in Y:\sup\{\langle x,y\rangle :x\in C\}\leq 0\}}$. [1]

## Geometry

In geometry, the polar set may also refer to a duality between points and planes. In particular, the polar set of a point ${\displaystyle x_{0}}$, given by the set of points ${\displaystyle x}$ satisfying ${\displaystyle \langle x,x_{0}\rangle =0}$ is its polar hyperplane, and the dual relationship for a hyperplane yields its pole. In convex geometry the polar set of a convex set containing the origin is defined similarly but without taking the absolute values of the inner product ${\displaystyle \langle x,y\rangle }$ in the definition, resulting in a different set.