Fenchel–Moreau theorem

The Fenchel–Moreau theorem states that a proper lower semicontinuous convex function on a suitable locally convex space equals its biconjugate, formed by taking its convex conjugate twice.

Convex conjugate: A function assigning to each dual vector the supremum of its pairing with points minus the original function’s value. Taking this transform twice produces the biconjugate in the theorem.
Lower semicontinuity: A function is lower semicontinuous when each strict superlevel set is open, equivalently its epigraph is closed. This regularity condition makes the function’s epigraph suitable for separation.
Fenchel duality: A framework relating a convex optimization problem to a dual problem built from conjugate functions. The theorem justifies recovering closed convex objectives from dual data.
Legendre transform: A transform relating a differentiable strictly convex function to a dual function through its gradient. It is a narrower classical counterpart to the more general convex conjugate setting.
Biconjugate: The convex conjugate of a function’s convex conjugate. The theorem identifies this second transform with the original function under its hypotheses.
Proper convex function: An extended-real-valued convex function that never equals negative infinity and is finite somewhere. Properness excludes degenerate cases in the standard theorem.
Legendre–Fenchel transform: The convex conjugate transform defined by taking a supremum over primal variables. It is the transform whose repeated application yields the theorem’s representation.
Jensen inequality: An inequality placing a convex function of an average below the average of its values. It characterizes convexity but does not recover a function from dual data.
Supporting hyperplane: A hyperplane that touches a convex set while leaving the set in one closed half-space. Separation supplies affine minorants that recover function values.
Locally convex topological vector space: A topological vector space whose topology is generated by a family of seminorms. The theorem uses its continuous dual and separation structure.

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Bipolar theoremCompared with: It extends the same dual-reconstruction pattern from sets to convex functions.