# Polar set

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

## Definitions

There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis.[1] In each case, the definition describes a duality between certain subsets of a dual pair of (topological) vector spaces ${\displaystyle (X,Y)}$.

### Geometric definition

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

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

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 ${\displaystyle x\in X}$ is the locus ${\displaystyle \{y:\langle y,x\rangle =0\}}$; the dual relationship for a hyperplane yields that hyperplane's polar point.[2]

Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.[3]

### Functional analytic-definition

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

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

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

Some authors include absolute values around the inner product; the two definitions coincide for circled sets.[1][3]

## Properties

• If ${\displaystyle A\subseteq B}$ then ${\displaystyle B^{\circ }\subseteq A^{\circ }}$
• An immediate corollary is that ${\displaystyle \bigcup _{i\in I}A_{i}^{\circ }\subseteq (\bigcap _{i\in I}A_{i})^{\circ }}$; equality necessarily holds only for finitely-many terms.
• For all ${\displaystyle \gamma \neq 0}$ : ${\displaystyle (\gamma A)^{\circ }={\frac {1}{\mid \gamma \mid }}A^{\circ }}$.
• ${\displaystyle (\bigcup _{i\in I}A_{i})^{\circ }=\bigcap _{i\in I}A_{i}^{\circ }}$.
• For a dual pair ${\displaystyle (X,Y)}$ ${\displaystyle A^{\circ }}$ is closed in ${\displaystyle Y}$ under the weak-*-topology on ${\displaystyle Y}$.[2]
• The bipolar ${\displaystyle A^{\circ \circ }}$ of a set ${\displaystyle A}$ is the closed convex hull of ${\displaystyle A\cup \{0\}}$, that is the smallest closed and convex set containing both ${\displaystyle A}$and ${\displaystyle 0}$.
• Similarly, the bidual cone of a cone ${\displaystyle A}$ is the closed conic hull of ${\displaystyle A}$.[4]
• For a closed convex cone ${\displaystyle C}$ in ${\displaystyle X}$, the dual cone is the polar of ${\displaystyle C}$; that is,
${\displaystyle C^{\circ }=\{y\in Y:\sup\{\langle x,y\rangle :x\in C\}\leq 0\}}$[1]

## References

1. Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 215. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.
2. Zălinescu, C. (2002). Convex Analysis in General Vector Spaces. River Edge, NJ: World Scientific. pp. 7–8. ISBN 978-9812380678.
3. Rockafellar, T.R. (1970). Convex Analysis. Princeton University. pp. 121–8. ISBN 978-0-691-01586-6.
4. Niculescu, C.P.; Persson, Lars-Erik (2018). Convex Functions and Their Applications. CMS Books in Mathematics. Cham, Switzerland: Springer. pp. 94–5, 134–5. doi:10.1007/978-3-319-78337-6. ISBN 978-3-319-78337-6.