On the pointwise maximum of convex functions

We study the conjugate of the maximum, $f \vee g$, of $f$ and $g$ when $f$ and $g$ are proper convex lower semicontinuous functions on a Banach space $E$. We show that $(f \vee g)^{**} = f^{**} \vee g^{**}$ on the bidual, $E^{**}$, of $E$ provided that $f$and $g$ satisfy the Attouch-Brézis constraint qualification, and we also derive formulae for $(f \vee g)^{*}$ and for the ``preconjugate'' of $f^{*}\vee g^{*}$.