- 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 ).
There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis. In each case, the definition describes a duality between certain subsets of a dual pair of (topological) vector spaces .
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.
Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.
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 ).
- 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 .
- The bipolar of a set is the closed convex hull of , that is the smallest closed and convex set containing both and .
- For a closed convex cone in , the dual cone is the polar of ; that is,
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