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

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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

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

References

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^ ^a ^b ^c 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. 103 (1). American Mathematical Society: 85–90. doi:10.2307/2047532. JSTOR 2047532.
^ Shozo Koshi; Naoto Komuro (1983). "A generalization of the Fenchel–Moreau theorem". Proc. Japan Acad. Ser. A Math. Sci.. 59 (5): 178–181.

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