Biconjugate
= Biconjugate
{title2=$f^{**}$}
= Fenchel biconjugate
{c}
{synonym}
The <biconjugate> of $f$ is the <convex conjugate> of its <convex conjugate>:
$$
f^{**}(x)=\sup_p\{\langle p,x\rangle-f^*(p)\}.
$$
It is the supremum of all <affine minorants>. For a proper function with an affine minorant, the <Fenchel-Moreau theorem> identifies it with the closed convex envelope; a proper <lower semicontinuous> <convex function> equals its biconjugate.