Polar set
- See also polar set (potential theory).
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 ).
Contents
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 .
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]}^{[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 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
- ^ ^{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 9812380671.
- ^ ^{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. Convex Functions and Their Applications. CMS Books in Mathematics. Cham, Switzerland: Springer. pp. 94–5, 134–5. ISBN 978-3-319-78337-6.
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