Abstract

The paper aims at creating a new insight into our perception of convexity by focusing on two fundamental problems: the coincidence of two functions (at least one being convex) upon an information on a dense set and the clarification of the relation between convexity and Fenchel subdifferential. Various results are established into these directions. Several examples are also illustrated showing that some rather unexpected situations can often occur.

Author information

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

Joël Benoist

Faculté des Sciences, Université de Limoges, 123 Av. Albert Thomas, 87060 Limoges, France
and: LACO, URA CNRS 1586

benoist@unilim.fr

Aris Daniilidis

Lab. de Mathématiques Appliquées, Université de Pau et des Pays de l'Adour, 64000 Pau, France
and: CNRS ERS 2055

aris.daniilidis@univ-pau.fr

Keywords

  • Convex function
  • coincidence
  • subdifferential

Mathematics Subject Classification

52A41, 49J52; 26E99

References

  1. M. Bachir, A. Daniilidis, J.-P. Penot: Lower subdifferentiability and integration, Set-Valued Anal. (to appear).Google Scholar
  2. J. Benoist, A. Daniilidis: Dual characterizations of relative continuity of convex functions, J. Austral. Math. Soc. 70 (2001) 211–223.https://doi.org/10.1017/S1446788700002615Google Scholar
  3. J. Borwein, Q. Zhu: Multifunctional and functional analytic techniques in nonsmooth analysis, in: Nonlinear Analysis, Differential Equations and Control (Montreal, QC, 1998), 61–157, NATO Sci. Ser. C, Math. Phys. Sci. 528, Kluwer Acad. Publ., Dordrecht (1999).https://doi.org/10.1007/978-94-011-4560-2_2Google Scholar
  4. J.-B. Hiriart-Urruty, C. Lemarechal: Convex Analysis and Minimization Algorithms I & II, Grundlehren der mathematischen Wissenschaften 306, Springer-Verlag, Berlin (1993).https://doi.org/10.1007/978-3-662-02796-7Google Scholar
  5. R. Phelps: Convex Functions, Monotone Operators and Differentiability, 2nd Edition, Springer-Verlag, Berlin (1991).Google Scholar
  6. R. Poliquin: Integration of subdifferentials of nonconvex functions, Nonlinear Anal. 17 (1991) 385–398.https://doi.org/10.1016/0362-546X(91)90078-FGoogle Scholar
  7. R. T. Rockafellar: Convex Analysis, Princeton Mathematical Series 28, Princeton, N.J. (1970).https://doi.org/10.1515/9781400873173Google Scholar
  8. L. Thibault, D. Zagrodny: Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995) 33–58.https://doi.org/10.1006/jmaa.1995.1003Google Scholar
  9. Z. Wu, J. Ye: Some results on integration of subdifferentials, Nonlinear Anal. 39 (2000) 955–976.https://doi.org/10.1016/S0362-546X(98)00263-6Google Scholar

Suggested citation

J. Benoist, A. Daniilidis. “Coincidence Theorems for Convex Functions.” Journal of Convex Analysis 9 (2002), No. 1, 259–268. https://doi.org/10.68381/jca09013

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Published by Heldermann Verlag, 2002. Rights now held by Banach Press.