A necessary and sufficient condition on the stability of the infimum of convex functions
Abstract
Let us say that a convex function f C[-∞,∞] on a convex set C⊂eq is infimum-stable if, for any sequence (fn) of convex functions fn C[-∞,∞] converging to f pointwise, one has ∈fC fn∈fC 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) in place of sequences (fn). This note is motivated by certain applications to stability of measures of risk/inequality in finance/economics.
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