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.^[1]^{[citation needed]} In each case, the definition describes a duality between certain subsets of a dual pair of (topological) vector spaces $(X,Y)$ .

Geometric definition

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.^[2]^{[citation needed]}

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 $A\subseteq X$ 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.^[1]^[3]

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 $Y$ .^[2]
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 $A$ .^[4]
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\}

^[1]

References

^ ^a ^b ^c 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.
^ ^a ^b Zălinescu, C. (2002). Convex Analysis in General Vector Spaces. River Edge, NJ: World Scientific. pp. 7–8. ISBN 978-9812380678.
^ ^a ^b Rockafellar, T.R. (1970). Convex Analysis. Princeton University. pp. 121–8. ISBN 978-0-691-01586-6.
^ 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.

[FctAnal-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.

[VecBook-2] Zălinescu, C. (2002). Convex Analysis in General Vector Spaces. River Edge, NJ: World Scientific. pp. 7–8. ISBN 978-9812380678.

[ConvAnal-3] Rockafellar, T.R. (1970). Convex Analysis. Princeton University. pp. 121–8. ISBN 978-0-691-01586-6.

[ConvFct-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.

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