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

TL;DR

This paper establishes a precise necessary and sufficient condition for the stability of the infimum of convex functions under pointwise convergence, with implications for risk and inequality measures in economics.

Contribution

It provides a simple, exact criterion characterizing infimum stability of convex functions, extending to Moore--Smith nets, with applications in finance and economics.

Findings

Characterizes infimum stability via a necessary and sufficient condition.

Extends the condition to Moore--Smith nets.

Motivated by applications in risk and inequality measures.

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