An Attempt of Characterization of Functions with Sharp Weakly Complete Epigraphs

We attempt to characterize the property, for a convex function f , "Epi f is sharp and weakly complete" by a property of the polar function. Some examples show that a result completely analogous to the J.L. Joly characterization of "Epi f is sharp and weakly locally compact" seems impossible. We use an old (1962) result of G. Choquet.