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A closed proper convex function equals its Fenchel biconjugate.
Fenchel-Moreau theorem: Let be a proper function, and let denote its biconjugate (defined using the Fenchel conjugate). Then is a closed convex function and satisfies
Moreover, pointwise if and only if is closed and convex.
This result justifies representing closed convex functions as suprema of affine minorants and is a structural cornerstone for convex duality.
A biconjugate of an extended-real-valued function is the function
where is the Fenchel conjugate of .
If is proper and has an affine minorant, then is the greatest lower-semicontinuous convex function bounded above by . In particular, pointwise. The Fenchel–Moreau theorem states that a proper function satisfies exactly when it is lower-semicontinuous and convex.
Here “lower-semicontinuous and convex” is also commonly called closed convex.
A Fenchel conjugate of an extended-real-valued function is the function defined by
where is the Euclidean inner product.
A closed convex function is an extended-real-valued function that is convex and has a closed epigraph
so that is a closed convex set in .
A primal-dual pair in convex optimization is a pair of problems built from convex objectives so that the dual objective gives a universal lower bound on the primal optimal value. A common (Fenchel) form starts from convex functions and and a linear map , and defines
where and are Fenchel conjugates and use infimum and supremum. The inequality (weak duality) follows from the Fenchel-Young inequality, and under suitable regularity conditions one has strong duality , often certified by subgradient conditions.