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 minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of som ...

, the Fenchel–Moreau theorem (named after

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

and

Jean Jacques Moreau Jean Jacques Moreau (31 July 1923 – 9 January 2014) was a French mathematician and mechanician. He normally published under the name J. J. Moreau. Moreau was born in Blaye. He received his doctorate in mathematics from the University of Paris, ...

) or Fenchel biconjugation theorem (or just biconjugation theorem) is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

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 set, polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions f ...

. It is used in

duality theory In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...

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 v ...

, 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 Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...

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, ro ...

, and convex function, #

f \equiv +\infty

, or #

f \equiv -\infty

References

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