A proper lower-semicontinuous convex functional on a real Banach space equals its biconjugate, with the canonical embedding in the bidual understood. One inequality follows from the Fenchel–Young inequality; separation of a point below the closed convex epigraph gives the opposite inequality. This is the closed-convex analogue of recovering a function from all its affine supporting functions.
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The Fenchel–Moreau theorem is a fundamental result in convex analysis that relates the concepts of convex conjugates and duality. It characterizes the relationship between a convex function and its conjugate. Let \( f : \mathbb{R}^n \to \mathbb{R} \) be a proper, convex, and lower semicontinuous function.