Bipolar theorem

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In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.[1]: 76–77 

Preliminaries[edit]

Suppose that is a topological vector space (TVS) with a continuous dual space and let for all and The convex hull of a set denoted by is the smallest convex set containing The convex balanced hull of a set is the smallest convex balanced set containing

The polar of a subset is defined to be:

while the prepolar of a subset is:
The bipolar of a subset often denoted by is the set

Statement in functional analysis[edit]

Let denote the weak topology on (that is, the weakest TVS topology on making all linear functionals in continuous).

The bipolar theorem:[2] The bipolar of a subset is equal to the -closure of the convex balanced hull of

Statement in convex analysis[edit]

The bipolar theorem:[1]: 54 [3] For any nonempty cone in some linear space the bipolar set is given by:

Special case[edit]

A subset is a nonempty closed convex cone if and only if when where denotes the positive dual cone of a set [3][4] Or more generally, if is a nonempty convex cone then the bipolar cone is given by

Relation to the Fenchel–Moreau theorem[edit]

Let

be the indicator function for a cone Then the convex conjugate,
is the support function for and Therefore, if and only if [1]: 54 [4]

See also[edit]

References[edit]

  1. ^ a b c Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 9780387295701.
  2. ^ Narici & Beckenstein 2011, pp. 225–273.
  3. ^ a b Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 9780521833783. Retrieved October 15, 2011.
  4. ^ a b Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–125. ISBN 9780691015866.

Bibliography[edit]