REGULARITY CONDITIONS FOR FORMULAE OF BICONJUGATE FUNCTIONS

Abstract

When the dual of a normed space $X$ is endowed with the weak$^*$ topology, the biconjugates of the proper convex lower semicontinuous functions defined on $X$ coincide with the functions themselves. This is not the case when $X^*$ is endowed with the strong topology. Working in the latter framework, we give formulae for the biconjugates of some functions that appear often in convex optimization, which hold provided the validity of some suitable regularity conditions. We also treat some special cases, rediscovering and improving recent results in the literature. Finally, we give a regularity condition that guarantees that the biconjugate of the supremum of a possibly infinite family of proper convex lower semicontinuous functions defined on a separated locally convex space coincides with the supremum of their biconjugates.