- See also polar set (potential theory).
Given a dual pair the polar set or polar of a subset of is a set in defined as
The bipolar of a subset of is the polar of . It is denoted and is a set in .
- is absolutely convex
- If then
- So , where equality of sets does not necessarily hold.
- For all :
- For a dual pair is closed in under the weak-*-topology on
- The bipolar of a set is the absolutely convex envelope of , that is the smallest absolutely convex set containing . If is already absolutely convex then .
- For a closed convex cone in , the polar cone is equivalent to the one-sided polar set for , given by
In geometry, the polar set may also refer to a duality between points and planes. In particular, the polar set of a point , given by the set of points satisfying is its polar hyperplane, and the dual relationship for a hyperplane yields its pole.
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Discussion of Polar Sets in Potential Theory: Ransford, Thomas: Potential Theory in the Complex Plane, London Mathematical Society Student Texts 28, CUP, 1995, pp. 55-58.
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