Lower conjugate
= Lower conjugate
{title2=$F_*(\tau)=\inf_e\{\tau:e-F(e)\}$}
The <lower conjugate> uses an infimum, unlike the supremum defining a <convex conjugate>. For any $F$, $F_*(\tau)=-(-F)^*(-\tau)$ and $F(e)\le\tau:e-F_*(\tau)$. The inequality is informative only when the <lower conjugate> is finite. If the infimum is attained uniquely and the dual is differentiable, $DF_*(\tau)$ is its minimizing argument. This sign convention is useful for upper comparison-energy bounds.