On the relations between two types of convergence for convex functions

Prox-regular functions in variational analysis

“…But then the functions f t , which epi-converge to f 0 , are convex on t −1 U . Since convex functions epi-converge on an open set O to a finite function if and only if they converge pointwise uniformly on every bounded subset of O (see Salinetti and Wets [33]), our claim about the second-order expansion of f is justified. Now that we know f is convex on a neighborhood of 0, we also know that f is subdifferentially continuous at 0 and, for (x, v) sufficiently near to (0, 0) in gph ∂f , the function ∆ x,v,t f is convex.…”

Graphical methods in first and second-order differentiability theory of integral functionals

“…Similarly, epi convergence requires the functions fm f to be lower semi-continuous in the norm topology, but this is a reasonable requirement even in the nonconvex case. We refer to [2,4,17,37,13] for a discussion of these notions.…”

Uniform Continuity on Bounded Sets and the Attouch-Wets Topology

“…With respect to convexity, convergence in this sense is stable with respect to duality, as established implicitly by Walkup and Wets [WW] and then by Wijsman [Wi]: If (An) is a sequence of closed convex sets in R" Kuratowski convergent to A , we have the convergence of the polar sequence (A°n) to A°. Attempts to obtain a suitable infinite dimensional generalization of this convergence notion have focused on the notion of Mosco convergence [Mol,Mo2,BB,BF,At,SW,So,Ts] and the associated Mosco topology [Be3,Be4]. Unfortunately, these ideas do not work well without reflexivity [BB].…”