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 $C\subseteq\mathbb{R}$ 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 $$\inf\limits_C f_n\to\inf\limits_C f.$$ 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.