Fenchel-Moreau theorem

Fenchel-Moreau theorem: Let $f:\mathbb{R}^n\to(-\infty,+\infty]$ be a proper function , and let $f^{**}$ denote its biconjugate (defined using the Fenchel conjugate ). Then $f^{**}$ is a closed convex function and satisfies

Moreover, $f=f^{**}$ pointwise if and only if $f$ is closed and convex.

This result justifies representing closed convex functions as suprema of affine minorants and is a structural cornerstone for convex duality .