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

References

  1. H. Attouch, H. Brezis: Duality for the sum of convex functions in general Banach spaces, in: Aspects of Mathematics and its Applications, North-Holland Mathematical Library Vol. 34, Elsevier, Amsterdam (1986) 125–133.https://doi.org/10.1016/S0924-6509(09)70252-1Google Scholar
  2. A. Brøndsted, R. T. Rockafellar: On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16/4 (1965) 605–611.https://doi.org/10.1090/S0002-9939-1965-0178103-8Google Scholar
  3. I. Ekeland: On the variational principle, J. Math. Analysis Appl. 47/2 (1974) 324–353.https://doi.org/10.1016/0022-247X(74)90025-0Google Scholar
  4. V. L. Klee: Extremal structure of convex sets. II, Math. Zeitschrift 69/1 (1958) 90–104.https://doi.org/10.1007/BF01187394Google Scholar
  5. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics Vol. 1364, Springer, Berlin (1989).https://doi.org/10.1007/978-3-662-21569-2Google Scholar
  6. J. Rainwater: Yet more on the differentiability of convex functions, Proc. Amer. Math. Soc. 103/3 (1988) 773–778.https://doi.org/10.1090/S0002-9939-1988-0947656-7Google Scholar
  7. S. M. Robinson: Regularity and stability for convex multivalued functions, Math. Oper. Res. 1/2 (1976) 130–143.https://doi.org/10.1287/moor.1.2.130Google Scholar
  8. R. T. Rockafellar: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970) 209–216.https://doi.org/10.2140/pjm.1970.33.209Google Scholar
  9. R. T. Rockafellar: Conjugate Duality and Optimization, BMS-NSF Regional Conference Series in Applied Mathematics Vol. 16, Society for Industrial and Applied Mathematics, Pittsburgh (1974).https://doi.org/10.1137/1.9781611970524.ch1Google Scholar

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.