Fenchel–Moreau theorem - Alchetron, The Free Social Encyclopedia

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 . 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 of theorem

Let ( X , τ ) be a Hausdorff locally convex space, for any extended real valued function f : X → R ∪ { ± ∞ } 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 ≡ + ∞ , or
f ≡ − ∞ .

References

Fenchel–Moreau theorem Wikipedia

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