Fenchel–Moreau theorem explained

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

Let

(X,\tau) be a Hausdorff locally convex space, for any extended real valued function f: X \to R \cup \{\pm inf ty\} it follows that f = f ^{**} if and only if one of the following is truef is a proper, lower semi-continuous, and convex function,f \equiv +inf ty , orf \equiv -inf ty .^{[1] [3] [4]}

Notes and References

1. Book: Borwein . Jonathan . Jonathan Borwein. Lewis . Adrian . Convex Analysis and Nonlinear Optimization: Theory and Examples. 2 . 2006 . Springer . 9780387295701. 76–77.
2. Book: Zălinescu, Constantin . Convex analysis in general vector spaces . World Scientific Publishing Co., Inc. . 981-238-067-1 . 1921556 . J . 2002 . River Edge, NJ . 75–79.
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 . 10.2307/2047532 . 2047532 . free .
4. A generalization of the Fenchel–Moreau theorem. Shozo Koshi. Naoto Komuro. Proc. Japan Acad. Ser. A Math. Sci.. 59. 5. 1983. 178–181.