Concave biconjugate
= Concave biconjugate
{title2=$F_{**}(e)=\inf_\tau\{\tau:e-F_*(\tau)\}$}
The <concave biconjugate> is an upper envelope of $F$, and equals $F$ for a proper closed <concave function> under the usual duality hypotheses. It equals $-(-F)^{**}$ in terms of the ordinary <biconjugate>. If $e=DF_*(\tau)$, differentiability and concavity of $F_*$ imply $F_*(\tau)+F_{**}(e)=\tau:e$. For a finite differentiable dual the equality follows because $\tau$ minimizes $s:e-F_*(s)$.