On the Relations between Two Types of Convergence for Convex Functions
چکیده
Theory and applications have shown that there are two important types of convergence for convex functions: pointwise convergence and convergence in a topology induced by the convergence of their epigraphs. We show that these two types of convergence are equivalent on the class of convex functions which are equi-lower semicontinuous. This turns out to be maximal classes of convex functions for which this equivalence can be obtained. We also indicate a number of implications of these results to the convergence of convex sets and the corresponding support functions and to the convergence of the i&ma of sequences of convex minimization problems.