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 In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformatio ...

. This is in contrast to the general property that for any function

f^ \leq f

. This can be seen as a generalization of the bipolar theorem. It is used in duality theory to prove

strong duality Strong duality is a condition in mathematical optimization in which the primal optimal objective and the dual optimal objective are equal. This is as opposed to weak duality (the primal problem has optimal value smaller than or equal to the dual ...

(via the

perturbation function In mathematical optimization, the perturbation function is any function which relates to primal and dual problems. The name comes from the fact that any such function defines a perturbation of the initial problem. In many cases this takes the form ...

Statement

Let

(X,\tau)

be a Hausdorff locally convex space, for any extended real valued function

f: X \to \mathbb \cup \

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 In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...

, #

f \equiv +\infty

, or #

f \equiv -\infty

References

{{DEFAULTSORT:Fenchel-Moreau theorem Convex analysis Theorems in analysis Theorems involving convexity