Polar set

In functional and convex analysis, related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space lying in the dual space . The bipolar of a subset is the polar of , but lies in (not ).

See also polar set (potential theory).

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 .

Geometric definition

The polar cone of a convex cone is the set

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 is the locus ; 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 is the set

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

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

Properties

  • If then
    • An immediate corollary is that ; equality necessarily holds only for finitely-many terms.
  • For all  : .
  • .
  • For a dual pair is closed in under the weak-*-topology on .[2]
  • The bipolar of a set is the closed convex hull of , that is the smallest closed and convex set containing both and .
    • Similarly, the bidual cone of a cone is the closed conic hull of .[4]
  • For a closed convex cone in , the dual cone is the polar of ; that is,
[1]

See also

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.
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