Lower conjugate ($F_{\ast }(\tau )={\inf }_e\{\tau :e-F(e)\}$)

The lower conjugate uses an infimum, unlike the supremum defining a convex conjugate. For any $F$, $F_{\ast }(\tau )=-(-F{)}^{\ast }(-\tau )$ and $F(e)\le \tau :e-F_{\ast }(\tau )$. The inequality is informative only when the lower conjugate is finite. If the infimum is attained uniquely and the dual is differentiable, $D{F}_{\ast }(\tau )$ is its minimizing argument. This sign convention is useful for upper comparison-energy bounds.

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 $-(-F{)}^{\ast \ast }$ in terms of the ordinary biconjugate. If $e=D{F}_{\ast }(\tau )$, differentiability and concavity of $F_{\ast }$ imply $F_{\ast }(\tau )+F_{\ast \ast }(e)=\tau :e$. For a finite differentiable dual the equality follows because $\tau$ minimizes $s:e-F_{\ast }(s)$.