Proposition 21.11.16E013E1966E44

Proof. For a closed proper $F$, finite-dimensional Fenchel duality gives \[F(x)=F^{**}(x)=\sup_p\{\langle p,x\rangle-F^*(p)\}.\] For every $p$ with $F^*(p)<+\infty$, the expression in braces is an affine minorant of $F$, by the definition of conjugacy. Conversely, every affine minorant $\langle p,x\rangle+c$ satisfies $c\leq-F^*(p)$. Thus the supremum is exactly that of all affine minorants. If $F\equiv+\infty$, arbitrary constant minorants already have supremum $+\infty$. If $F\equiv-\infty$, there are no finite affine minorants and their supremum is $-\infty$.