Bipolar theorem

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.

For any nonempty set ${\displaystyle C\subset X}$ in some linear space ${\displaystyle X}$, then the bipolar cone ${\displaystyle C^{oo}=(C^{o})^{o}}$ is given by

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