# 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 .[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

- Borwein, Jonathan; Lewis, Adrian (2006).
*Convex Analysis and Nonlinear Optimization: Theory and Examples*(2 ed.). Springer. pp. 76–77. ISBN 9780387295701. - 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. - 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. - Shozo Koshi; Naoto Komuro (1983). "A generalization of the Fenchel–Moreau theorem".
*Proc. Japan Acad. Ser. A Math. Sci.*.**59**(5): 178–181.

This article is issued from
Wikipedia.
The text is licensed under Creative
Commons - Attribution - Sharealike.
Additional terms may apply for the media files.