Bipolar theorem
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In mathematics, the bipolar theorem is a theorem in convex analysis which provides 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
Statement of theorem
For any nonempty set in some linear space
, then the bipolar cone
is given by
where denotes the convex hull.[1]:54[2]
Special case
is a nonempty closed convex cone if and only if
when
, where
denotes the positive dual cone.[2][3]
Or more generally, if is a nonempty convex cone then the bipolar cone is given by
Relation to Fenchel–Moreau theorem
If is the indicator function for a cone
. Then the convex conjugate
is the support function for
, and
. Therefore
if and only if
.[1]:54[3]