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Journal of Convex Analysis 26 (2019), No. 1, 077--087
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.


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.

Keywords: Convex functions, minimization, stability, convergence, Legendre-Fenchel transform.

MSC: 26A51, 90C25; 49J45, 49K05, 49K30

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