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 ∗ ∗ ≤ f {\displaystyle f^{**}\leq f} . This can be seen as a generalization of the bipolar theorem. It is used in duality theory to prove strong duality (via the perturbation function).
Statement Let ( X , τ ) {\displaystyle (X,\tau )} be a Hausdorff locally convex space, for any extended real valued function f : X → R ∪ { ± ∞ } {\displaystyle f:X\to \mathbb {R} \cup \{\pm \infty \}} it follows that f = f ∗ ∗ {\displaystyle f=f^{**}} if and only if one of the following is true
f {\displaystyle f} is a proper, lower semi-continuous, and convex function,
f ≡ + ∞ {\displaystyle f\equiv +\infty } , or
f ≡ − ∞ {\displaystyle f\equiv -\infty } .
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