Abstract

We study the existence of residual sets of certain classes of convex functions on a normed linear space. Such sets play an important role in many optimization algorithms for which elements of a certain class can be used as good approximants of given convex functions and thus the abundance of such elements is crucial for the aforementioned algorithms.

Author information

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

Simeon Reich

Dept. of Mathematics, The Technion – Israel Institute of Technology, Haifa, Israel

sreich@technion.ac.il

Alexander J. Zaslavski

Dept. of Mathematics, The Technion – Israel Institute of Technology, Haifa, Israel

ajzasl@technion.ac.il

Keywords

  • Baire category
  • convex function
  • local uniform convexity
  • lower semi-continuity
  • normed space
  • residual set
  • strict convexity
  • uniformity

Mathematics Subject Classification

46N10, 54E50, 54E52

References

  1. G. Beer: Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht (1993).https://doi.org/10.1007/978-94-015-8149-3Google Scholar
  2. J. M. Borwein, J. D. Vanderwerff: Convex Functions: Constructions, Characterizations and Counterexamples, Cambridge University Press, Cambridge (2010).https://doi.org/10.1017/CBO9781139087322Google Scholar
  3. D. Butnariu, S. Reich, A. J. Zaslavski: There are many totally convex functions, J. Convex Analysis 13 (2006) 623–632.https://doi.org/10.68381/jca13048Google Scholar
  4. J. B. Conway: A Course in Functional Analysis, 2nd ed., Springer, New York (1990).Google Scholar
  5. J. L. Kelley: General Topology, D. Van Nostrand, New York (1955).Google Scholar
  6. A. R. Lovaglia: Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78 (1955) 225–238.https://doi.org/10.1090/S0002-9947-1955-0066558-7Google Scholar
  7. E. Michael: A note on completely metrizable spaces, Proc. Amer. Math. Soc. 96 (1986) 513–522.https://doi.org/10.1090/S0002-9939-1986-0822451-6Google Scholar
  8. S. Reich, A. J. Zaslavski: Genericity in Nonlinear Analysis, Springer, New York (2014).https://doi.org/10.1007/978-1-4614-9533-8Google Scholar
  9. J. Vandewerff: On the residuality of certain classes of convex functions, Pure Appl. Funct. Analysis 5 (2020) 791–806.Google Scholar

Suggested citation

K. Barshad, S. Reich, A. J. Zaslavski. “Residuality Properties of Certain Classes of Convex Functions on Normed Linear Spaces.” Journal of Convex Analysis 29 (2022), No. 3, 795–806. https://doi.org/10.68381/jca29046

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Published by Heldermann Verlag, 2022. Rights now held by Banach Press.