# Fenchel–Moreau theorem

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In convex analysis, the **Fenchel–Moreau theorem** (named after Werner Fenchel and Jean Jacques Moreau) or **Fenchel biconjugation theorem** (or just **biconjugation theorem**) is a theorem which gives necessary and sufficient conditions for a function to be equal to its biconjugate. This is in contrast to the general property that for any function .^{[1]}^{[2]} This can be seen as a generalization of the bipolar theorem.^{[1]} It is used in duality theory to prove strong duality (via the perturbation function).

## Statement of theorem

Let be a Hausdorff locally convex space, for any extended real valued function it follows that if and only if one of the following is true

- is a proper, lower semi-continuous, and convex function,
- , or
- .
^{[1]}^{[3]}^{[4]}

## References

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