Journal of Convex Analysis 33 (2026), No. 3&4, 1095--1104
Copyright Heldermann Verlag 2026
On the Weak* Convergence of Gâteaux Derivatives of Upper Envelopes
Dariusz Zagrodny
Faculty of Mathematics and Computer Science, University of Lodz, Poland, Lodz, Poland
dariusz.zagrodny@wmii.uni.lodz.pl
[Abstract-pdf]
For convex functions on Banach space endowed with a uniformly G\^ateaux differentiable norm two observations are presented: first, the Moreau envelope of a proper lower semicontinuous convex function is G\^ateaux differentiable; second, if the Moreau envelopes of a sequence of lower semicontinuous convex functions $\{f_n\}_{n=1}^{\infty}$ create a pointwise convergent sequence, then G\^ateaux derivatives of the envelopes (computed at a given point) are weakly$^*$ convergent.
Keywords: Subdifferentials, convex functions, Attouch's theorem, UG-smooth norm, Gateaux differentiability of the Moreau envelope, weak convergence of subdifferentials.
MSC: 49J52.
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