A necessary and sufficient condition on the stability of the infimum of convex functions

Abstract

Let us say that a convex function f\colon C\to[-\infty,\infty] on a convex set C\subseteq\R is infimum-stable if, for any sequence (f_n) of convex functions f_n\colon C\to[-\infty,\infty] converging to f pointwise, one has \inf_C f_n\to\inf_C f. A simple necessary and sufficient condition for a convex function to be infimum-stable is given. The same condition remains necessary and sufficient if one uses Moore--Smith nets (f_ν) in place of sequences (f_n). This note is motivated by certain applications to stability of measures of risk/inequality in finance/economics.

8 pages. Version 2: two references added; discussion expanded, including now Example 6. Version 3: discussion further expanded, including now a 5-line proof in the case when f and f_n are real-valued. Version 4: Theorem 9 is added about the equivalence of the sequential and topological variants of the infimum-stability