Abstract

We compare the Fenchel-Moreau second conjugates associated to an arbitrary coupling function φ:X×W→R‾=[−∞,+∞]\varphi:X\times W\rightarrow \overline{R}=[-\infty,+\infty ] between two sets XX and WW with the second level set conjugates associated to the same coupling. For a coupling φ:Rn×Rn→R=(−∞,+∞)\varphi:R^{n}\times R^{n}\rightarrow R=(-\infty,+\infty ) that is additively homogeneous in one (or both) of the variables we also compare the first conjugates associated to the same coupling. We give an application to the ``min-type'' coupling function arising in the study of topical functions.

Author information

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

Juan-Enrique Martínez-Legaz

Dep. d'Economia i d'Història Econòmica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

Ivan Singer

Institute of Mathematics, P.O. Box 1-764, 70700 Bucharest, Romania

Keywords

  • Generalized conjugation
  • topical functions
  • hull operators

Mathematics Subject Classification

49N15, 26B25, 52A01

References

  1. J.-P. Crouzeix: Contributions à l’Étude des Fonctions Quasi-Convexes, Thèse, Univ. de Clermont (1977).Google Scholar
  2. L. J. Lau: Duality and the structure of utility functions, J. Econ. Theory 1 (1970) 374–396.https://doi.org/10.1016/0022-0531(69)90024-6Google Scholar
  3. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, Amsterdam (1976).Google Scholar
  4. J. E. Martínez-Legaz: Generalized convex duality and its economic applications, in: Handbook of Generalized Convexity and Generalized Monotonicity, N. Hadjisavvas et al. (ed.), Nonconvex Optim. Appl. 76, Springer, New York (2005) 237–292.https://doi.org/10.1007/0-387-23393-8_6Google Scholar
  5. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  6. A. M. Rubinov, I. Singer: Topical and sub-topical functions, downward sets and abstract convexity, Optimization 50 (2001) 307–351.https://doi.org/10.1080/02331930108844567Google Scholar
  7. I. Singer: Conjugate functionals and level sets, Nonlinear Anal., Theory Methods Appl. 8 (1984) 313–320.https://doi.org/10.1016/0362-546X(84)90033-6Google Scholar
  8. I. Singer: Abstract Convex Analysis, Wiley, New York (1997).Google Scholar
  9. I. Singer: Further applications of the additive min-type coupling function, Optimization 51 (2002) 471–485.https://doi.org/10.1080/0233193021000004949Google Scholar
  10. M. Volle: Conjugaison par tranches, Ann. Mat. Pura Appl., IV. Ser. 139 (1985) 279–311.https://doi.org/10.1007/BF01766858Google Scholar

Suggested citation

J.-E. Martínez-Legaz, I. Singer. “Comparing Fenchel-Moreau Conjugates with Level Set Conjugates.” Journal of Convex Analysis 15 (2008), No. 2, 285–297. https://doi.org/10.68381/jca15020

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.