Fenchel–Moreau theorem

Remarks

For every $f$, the function $f^{**}$ is a supremum of continuous affine functions, so it is convex and lower semicontinuous, and the Fenchel–Young inequality gives $f^{**} \le f$. If $f$ takes the value $-\infty$ somewhere, then $f^*$ is identically $+\infty$ and $f^{**}$ is identically $-\infty$, so $f^{**} = f$ forces case 3. Otherwise $f^{**} = f$ forces $f$ to be convex and lower semicontinuous, which is case 1 or case 2. For the "if" direction, take $x_0$ and a real $r < f(x_0)$. The point $(x_0, r)$ lies outside the closed convex epigraph of $f$, and the Hahn–Banach separation theorem gives a continuous affine minorant of $f$ with value above $r$ at $x_0$. Every such minorant is below $f^{**}$, so $f^{**}(x_0) \ge f(x_0)$.

Example: $f(x) = \abs{x}$ on $\R$ has $f^* = 0$ on $[-1, 1]$ and $+\infty$ elsewhere, and $f^{**}(x) = \sup_{\abs{y} \le 1} xy = \abs{x}$. Lower semicontinuity is necessary: for $f = 0$ on $(0, 1)$ and $f = +\infty$ elsewhere on $\R$, the function is convex and proper, but $f^{**}$ is $0$ on the closed interval $[0, 1]$. Convexity is necessary: for $f(x) = \min\big((x - 1)^2, (x + 1)^2\big)$, the biconjugate is $0$ on $[-1, 1]$ and agrees with $f$ outside.

Going deeper

For a student

The theorem is the basis of duality in convex optimization. A primal problem and its dual are related through conjugates, and the theorem says that dualizing twice gives back the primal problem. It also shows that $f \mapsto f^*$ is a bijection between proper convex lower semicontinuous functions on $X$ and on the dual side, when $X$ is reflexive. It extends the involution property of the classical Legendre transformation to nonsmooth functions. The same statement holds in every Hausdorff locally convex space.