Journal of Convex Analysis 33 (2026), No. 3&4, 1013--1048
Copyright Heldermann Verlag 2026
Convergence of Subdifferentials versus Convergence of Slopes
Pedro Pérez-Aros
Dep. de Ingeniería Matemática, Centro de Modelamiento Matemático, Universidad de Chile, Santiago, Chile
pperez@dim.uchile.cl
Lionel Thibault
Université de Montpellier, Institut Montpelliérain A. Grothendieck, Montpellier, France
lionel.thibault@umontpellier.fr
Dariusz Zagrodny
Faculty of Mathematics and Computer Science, University of Lodz, Poland, Lodz, Poland
dariusz.zagrodny@wmii.uni.lodz.pl
Given a sequence of lower semicontinuous proper convex functions on a Banach space, the graph convergence of their subdifferentials as well as the epigraphical convergence of their slopes are investigated. Several (counter-)examples showing that equivalence between these convergences is uncommon in infinite dimension are provided. However, under suitable additional conditions it is possible to establish some relations of this kind. Herein, under a compactness type assumption it is shown that bounded mixed convergence of subdifferentials of the sequence of functions is equivalent to the convergence of subdifferentials in the sense of Painlevé-Kuratowski, which turns to be equivalent to the epigraphical convergence of slopes of the functions in separable Banach spaces.
Keywords: Convex function, slope, convergence of slopes, renorming, convergence of subdifferentials, bounded mixed way convergence of sets, perfect square criterion, maximality, compactness.
MSC: 49J52; 47J22, 58E30.
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