Abstract

We give an example of a convex, finite and lower semicontinuous function whose subdifferential is everywhere empty. This is possible since the function is defined on an incomplete normed space. The function serves as a universal counterexample to various statements in convex analysis in which completeness is required.

Author information

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

Gerd Wachsmuth

Institute of Mathematics, Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany

wachsmuth@b-tu.de

Keywords

  • Subdifferential
  • incomplete space
  • Fenchel duality
  • convex sum rule

Mathematics Subject Classification

46N10, 90C25

Suggested citation

G. Wachsmuth. “A Convex, Finite and Lower Semicontinuous Function with Empty Subdifferential.” Journal of Convex Analysis 32 (2025), No. 3, 877–882. https://doi.org/10.68381/jca32044

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press.