# Fenchel–Moreau theorem

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 $f^{**}\leq f$ . This can be seen as a generalization of the bipolar theorem. It is used in duality theory to prove strong duality (via the perturbation function).

## Statement of theorem

Let $(X,\tau )$ be a Hausdorff locally convex space, for any extended real valued function $f:X\to \mathbb {R} \cup \{\pm \infty \}$ it follows that $f=f^{**}$ if and only if one of the following is true

1. $f$ is a proper, lower semi-continuous, and convex function,
2. $f\equiv +\infty$ , or
3. $f\equiv -\infty$ .
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