convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, convex minimization, a subdomain of optimization (mathematics), optimization theor ...

, the Fenchel–Moreau theorem (named after

Werner Fenchel Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a German-Danish mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear opti ...

and Jean Jacques Moreau) or Fenchel biconjugation theorem (or just biconjugation theorem) is a

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

which gives

necessary and sufficient conditions In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

for a function to be equal to its biconjugate. 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 In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone ...

. It is used in

duality theory In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involution ...

to prove strong duality (via the perturbation function).

Statement

Let

(X,\tau)

be a Hausdorff

locally convex space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...

, 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 In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, r ...

, and

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

, #

f \equiv +\infty

, or #

f \equiv -\infty

References

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