Fenchel–Moreau theorem


In convex analysis, the Fenchel–Moreau theorem or Fenchel 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. This can be seen as a generalization of the bipolar theorem. It is used in duality theory to prove strong duality.

Statement

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
  1. is a proper, lower semi-continuous, and convex function,
  2. , or
  3. .