DOI: 10.68381/jca26005 ISSN: 0944-6532
A Necessary and Sufficient Condition on the Stability of the Infimum of Convex Functions
Iosif Pinelis
Let us say that a convex function
f\colon C\to[-\infty,\infty]
f
:
C
→
[
−
∞
,
∞
]
on a convex set
C\subseteq\mathbb{R}
C
⊆
R
is infimum-stable if, for any sequence
(f_n)
(
f
n
)
of convex functions
f_n\colon C\to[-\infty,\infty]
f
n
:
C
→
[
−
∞
,
∞
]
converging to
f
f
pointwise, one has
\inf\limits_C f_n\to\inf\limits_C f.
inf
C
f
n
→
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)
(
f
ν
)
in place of sequences
(f_n)
(
f
n
)
. This note is motivated by certain applications to stability of measures of risk/inequality in finance/economics.