Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Pedro Pérez-Aros

Dep. de Ingeniería Matemática, Centro de Modelamiento Matemático, Universidad de Chile, Santiago, Chile
and: CNRS UMI 2807, Santiago, Chile

pperez@dim.uchile.cl

Lionel Thibault

Université de Montpellier, Institut Montpelliérain A. Grothendieck, Montpellier, France
and: UMR 5149 CNRS, Montpellier, France

lionel.thibault@umontpellier.fr

Keywords

  • Convex function
  • slope
  • convergence of slopes
  • renorming
  • convergence of subdifferentials
  • bounded mixed way convergence of sets
  • perfect square criterion
  • maximality
  • compactness

Mathematics Subject Classification

49J52; 47J22, 58E30

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Suggested citation

P. Pérez-Aros, L. Thibault, D. Zagrodny. “Convergence of Subdifferentials versus Convergence of Slopes.” Journal of Convex Analysis 33 (2026), No. 3&4, 1013–1048. https://doi.org/10.68381/jca33054

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.