Fenchel–Moreau theorem

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

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 $f^{**} \leq f$.^[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 $(X,\tau)$ 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. $f \equiv -\infty$.^[1][3][4]

References

1. Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. pp. 76–77. ISBN 9780387295701.
2. Zălinescu, Constantin (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc. pp. 75–79. ISBN 981-238-067-1. MR 1921556.
3. Hang-Chin Lai; Lai-Jui Lin (May 1988). "The Fenchel-Moreau Theorem for Set Functions". Proceedings of the American Mathematical Society (American Mathematical Society) 103 (1): 85–90. doi:10.2307/2047532.
4. Shozo Koshi; Naoto Komuro (1983). "A generalization of the Fenchel–Moreau theorem". Proc. Japan Acad. Ser. A Math. Sci. 59 (5): 178–181.