Abstract

We show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems, concerning extendability of continuous convex functions from a closed subspace of a normed linear space, complement recent results of J. Borwein, V. Montesinos and J. Vanderwerff.

Author information

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

Libor Veselý

Dip. di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano, Italy

vesely@mat.unimi.it

Luděk Zajíček

Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8, Czech Republic

zajicek@karlin.mff.cuni.cz

Keywords

  • d.c. function
  • d.c. mapping
  • delta-convex mapping
  • convex function
  • extension
  • Banach space

Mathematics Subject Classification

52A41; 26B25, 46B99

References

  1. Y. Benyamini, J. Lindenstrauss: Geometric Nonlinear Functional Analysis. Volume 1, AMS Colloquium Publications 48, American Mathematical Society, Providence (2000).https://doi.org/10.1090/coll/048Google Scholar
  2. J. Borwein, S. Fitzpatrick, J. Vanderwerff: Examples of convex functions and classifications of normed spaces, J. Convex Analysis 1 (1994) 61–73.https://doi.org/10.68381/jca01004Google Scholar
  3. J. Borwein, V. Montesinos, J. Vanderwerff: Boundedness, differentiability and extensions of convex functions, J. Convex Analysis 13 (2006) 587–602.https://doi.org/10.68381/jca13046Google Scholar
  4. J. Borwein, J. Vanderwerff: On the continuity of biconjugate convex functions, Proc. Amer. Math. Soc. 130 (2002) 1797–1803.https://doi.org/10.1090/S0002-9939-01-06249-9Google Scholar
  5. J. Duda, L. Veselý, L. Zajíček: On d.c. functions and mappings, Atti Semin. Mat. Fis. Univ. Modena 51 (2003) 111–138.Google Scholar
  6. P. Hartman: On functions representable as a difference of convex functions, Pacific J. Math. 9 (1959) 707–713.https://doi.org/10.2140/pjm.1959.9.707Google Scholar
  7. F. A. Valentine: Convex Sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York (1964).Google Scholar
  8. L. Veselý, L. Zajíček: Delta-convex mappings between Banach spaces and applications, Diss. Math. 289 (1989) 1–52.Google Scholar
  9. L. Veselý, L. Zajíček: On compositions of d.c. functions and mappings, J. Convex Analysis 16 (2009) 423–439.https://doi.org/10.68381/jca16023Google Scholar

Suggested citation

L. Veselý, L. Zajíček. “On Extensions of D. C. Functions and Convex Functions.” Journal of Convex Analysis 17 (2010), No. 2, 427–440. https://doi.org/10.68381/jca17030

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.