On the pointwise maximum of convex functions

Source abstract

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