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 ${\displaystyle f^{**}\leq f}$.^[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).

A function that is not lower semi-continuous. By the Fenchel-Moreau theorem, this function is not equal to its biconjugate.

Statement

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

1. ${\displaystyle f}$ is a proper, lower semi-continuous, and convex function,
2. ${\displaystyle f\equiv +\infty }$, or
3. ${\displaystyle f\equiv -\infty }$.^[1][3][4]