Abstract

We survey various boundedness, differentiability and extendibility properties of convex functions, and how they are related to sequential convergence with respect to various topologies in the dual space. It is also shown that if X/Y is separable then every continuous convex function on Y can be extended to a continuous convex function on X.

Author information

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

Jonathan Borwein

Faculty of Computer Science, Dalhousie University, Halifax, N.S., Canada B3H 1W5

jborwein@cs.dal.ca

Vicente Montesinos

Instituto de Matemática Pura y Aplicada, Universidad Politécnica, C/Vera s/n, 46022 Valencia, Spain

vmontesinos@mat.upv.es

Jon D. Vanderwerff

Dept. of Mathematics, La Sierra University, Riverside, CA 92515, U.S.A.

jvanderw@lasierra.edu

Keywords

  • Convex function
  • Schur property
  • Dunford-Pettis property
  • Grothendieck property
  • extensions

Mathematics Subject Classification

52A41; 46G05, 46N10, 49J50, 90C25

References

  1. H. H. Bauschke, J. M. Borwein, P. L. Combettes: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001) 615–647.https://doi.org/10.1142/S0219199701000524Google Scholar
  2. J. M. Borwein: Asplund spaces are ‘sequential reflexive’, University of Waterloo Technical Report (1991).Google Scholar
  3. J. M. Borwein, M. Fabian: On convex functions having points of Gateaux differentiability which are not points of Fréchet differentiability, Can. J. Math. 45 (1993) 1121–1134.https://doi.org/10.4153/CJM-1993-062-8Google Scholar
  4. J. M. Borwein, M. Fabian, J. Vanderwerff: Characterizations of Banach spaces via convex and other locally Lipschitz functions, Acta Math. Vietnam. 22 (1997) 53–69.Google Scholar
  5. J. M. Borwein, S. Fitzpatrick, J. Vanderwerff: Examples of convex functions and classifications of normed spaces, J. Convex Analysis 1 (1994) 10–18.https://doi.org/10.68381/jca01004Google Scholar
  6. J. M. Borwein, J. Vanderwerff: Epigraphical and uniform convergence of convex functions, Trans. Amer. Math. Soc. 348 (1996) 1617–1631.https://doi.org/10.1090/S0002-9947-96-01581-4Google Scholar
  7. J. M. Borwein, J. Vanderwerff: Convex functions on Banach spaces not containing ℓ1, Can. Math. Bull. 40 (1997) 10–18.https://doi.org/10.4153/CMB-1997-002-1Google Scholar
  8. J. M. Borwein, J. Vanderwerff: On the continuity of biconjugate convex functions, Proc. Amer. Math. Soc. 130 (2001) 1797–1803.https://doi.org/10.1090/S0002-9939-01-06249-9Google Scholar
  9. M. D. Contreras, R. Payá: On upper semicontinuity of the duality mapping, Proc. Amer. Math. Soc. 121 (1994) 451–459.https://doi.org/10.1090/S0002-9939-1994-1215199-4Google Scholar
  10. R. Deville, G. Godefroy, V. Zizler: Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific & Technical, Harlow (1993).Google Scholar
  11. J. Diestel: Sequences and Series in Banach Spaces, Graduate Texts in Mathematics 92, Springer, Berlin (1984).https://doi.org/10.1007/978-1-4612-5200-9Google Scholar
  12. J. Diestel, J. J. Uhl: Vector Measures, AMS Mathematical Surveys 15, American Mathematical Society, Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  13. M. Fabian, P. Habala, P. Hájek, V. Montesinos, J. Pelant, V. Zizler: Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics 8, Springer, New York (2001).https://doi.org/10.1007/978-1-4757-3480-5Google Scholar
  14. C. Franchetti, R. Payá: Banach spaces with strongly subdifferentiable norm, Boll. Unione Mat. Ital., VII. Ser., B 7 (1993) 45–70.Google Scholar
  15. G. Godefroy: Prolongement de fonctions convexes définies sur un espace de Banach E au bidual E″, C. R. Acad. Sci., Paris, Sér. I 292 (1981) 371–374.Google Scholar
  16. G. Godefroy, V. Montesinos, V. Zizler: Strong subdifferentiability of norms and geometry of Banach spaces, Commentat. Math. Univ. Carol. 36 (1995) 493–502.Google Scholar
  17. R. Haydon: A non-reflexive Grothendieck space that does not contain ℓ∞, Israel J. Math. 40 (1981) 65–73.https://doi.org/10.1007/BF02761818Google Scholar
  18. R. B. Holmes: Geometric Functional Analysis and its Applications, Graduate Texts in Mathematics 24, Springer, New York (1975).https://doi.org/10.1007/978-1-4684-9369-6Google Scholar
  19. W. B. Johnson, J. Lindenstrauss: Extension of bounded linear operator, in: Handbook of the Geometry of Banach Spaces I, W. B. Johnson, J. Lindenstrauss (eds.), Elsevier, Amsterdam (2001) 1–84.Google Scholar
  20. J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces I. Sequence spaces, Springer, Berlin (1977).https://doi.org/10.1007/978-3-642-66557-8Google Scholar
  21. P. Ørno: On J. Borwein’s concept of sequentially reflexive Banach spaces, Banach Space Bulletin Board (1991).Google Scholar
  22. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, 2nd Ed., Lecture Notes in Mathematics 1364, Springer, Berlin (1993).Google Scholar
  23. A. N. Pličko: On projective resolutions of the identity operator and Markuševič bases, Sov. Math., Dokl. 25 (1982) 386–389.Google Scholar
  24. H. P. Rosenthal: On complemented and quasi-complemented subspaces of quotients of C(S) for Stonian S, Proc. Natl. Acad. Sci. USA 60 (1968) 1165–1169.https://doi.org/10.1073/pnas.60.4.1165Google Scholar
  25. H. P. Rosenthal: On injective Banach spaces and the spaces L∞ (µ) for finite measures µ, Acta. Math. 124 (1970) 205–248.https://doi.org/10.1007/BF02394572Google Scholar
  26. H. P. Rosenthal: The complete separable extension property, J. Oper. Theory 43 (2000) 329–374.Google Scholar
  27. H. H. Schaefer: Banach Lattices and Positive Operators, Springer, Berlin (1994).Google Scholar
  28. S. Simons: Maximal monotone multifunctions of Brønsted-Rockafellar type, Set-Valued Anal. 7 (1999) 255–294.https://doi.org/10.1023/A:1008779514159Google Scholar
  29. M. Valdivia: Fréchet spaces with no subspaces isomorphic to ℓ1, Math. Jap. 38 (1993) 397–411.Google Scholar
  30. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar
  31. M. Zippin: Extension of bounded linear operators, in: Handbook of the Geometry of Banach Spaces II, W. B. Johnson, J. Lindenstrauss (eds.), North-Holland, Amsterdam (2003) 1703–1741..https://doi.org/10.1016/S1874-5849(03)80047-5Google Scholar
  32. V. E. Zizler: Smooth extensions of norms and complementability of subspaces, Arch. Math. 53 (1989) 585–589.https://doi.org/10.1007/BF01199818Google Scholar

Suggested citation

J. Borwein, V. Montesinos, J. D. Vanderwerff. “Boundedness, Differentiability and Extensions of Convex Functions.” Journal of Convex Analysis 13 (2006), No. 3/4, 587–602. https://doi.org/10.68381/jca13046

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.