
Journal of Convex Analysis 26 (2019), No. 1, 077087 Copyright Heldermann Verlag 2019 A Necessary and Sufficient Condition on the Stability of the Infimum of Convex Functions Iosif Pinelis Dept. of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, U.S.A. ipinelis@mtu.edu [Abstractpdf] Let us say that a convex function $f\colon C\to[\infty,\infty]$ on a convex set $C\subseteq\mathbb{R}$ is infimumstable 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 infimumstable is given. The same condition remains necessary and sufficient if one uses MooreSmith 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. Keywords: Convex functions, minimization, stability, convergence, LegendreFenchel transform. MSC: 26A51, 90C25; 49J45, 49K05, 49K30 [ Fulltextpdf (103 KB)] for subscribers only. 