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

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_\nu) in place of sequences (f_n). This note is motivated by certain applications to stability of measures of risk/inequality in finance/economics.