Skip to main content
Log in

An Unsolved Problem of Fenchel

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The Fenchel problem of level sets is solved under the conditions that theboundaries of the nested family of convex sets in Rn>+1 aregiven by C3 n-dimensional differentiable manifolds and theconvex sets determine an open or closed convex set inRn+1.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arrow, K.J. and Enthoven, A.C., Quasi-concave programming, Econometrica 29(1961) 779–800.

    Google Scholar 

  2. Avriel, M., Diewert, W. E., Schaible, S. and Zang, I., Generalized Concavity, Plenum Press, New York, London, 1988.

    Google Scholar 

  3. Crouzeix, J.P., Contributions a l’étude des fonctions quasiconvexes, Thèse, Université de Clermont-Ferrand, 1977.

  4. Crouxeix, J.P., On second order conditions for quasiconvexity, Mathematical Programming 18 (1980) 349–352.

    Google Scholar 

  5. Debreu, G., Representation of a Preference Ordering by a Numerical Function, in: Decision Processes, Thrall, Coombs and Davis (eds.), John-Wiley and Sons, 1954.

  6. Debreu, G., Least concave utility functions, Journal of Mathematical Economics 3(1976) 121–129.

    Google Scholar 

  7. de Finetti, B., Sulle stratificazioni convesse, Annali di Matematica Pura ed Applicata 30(1949) 173–183.

    Google Scholar 

  8. Eisenhart, L. P., Riemannian Geometry, Princeton University Press, Princeton, 1964.

    Google Scholar 

  9. Fenchel, W., Convex Cones, Sets and Functions, (mimeographed lecture notes), Princeton University Press, Princeton, New Jersey, 1953.

    Google Scholar 

  10. Fenchel, W., Über konvexe Funktionen mit vorgeschriebenen Niveaumannigfaltigkeiten, Mathematische Zeitschrift 63(1956) 496–506.

    Google Scholar 

  11. Kannai, Y., Concavifiability and constructions of concave utility functions, Journal of Mathematical Economics 4(1977) 1–56.

    Google Scholar 

  12. Kannai, Y., Concave utility functions–existence, constructions and cardinality, in: Generalized concavity in optimization and economics, (eds.): S. Schaible and Ziemba, W.T., Academic Press, New York (1981) 543–611.

    Google Scholar 

  13. Komlósi, S., Second order characterization of pseudoconvex and strictly pseudoconvex functions in terms of quasi-Hessians, in: Contributions to the Theory of Optimization, (ed.): F. Forgó, Department of Mathematics, Karl Marx University of Economics, Budapest (1983) 19–46.

    Google Scholar 

  14. Rapcsák, T., On pseudolinear functions, European Journal of Operational Research 50(1991a) 353–360.

    Google Scholar 

  15. Rapcsák, T., Geodesic convexity in nonlinear optimization, Journal of Optimization Theory and Applications 69(1991b) 169–183.

    Google Scholar 

  16. Rapcsák, T., On the connectedness of the solution set to nonlinear complementarity systems, Journal of Optimization Theory and Applications 81(1994) 619–631.

    Google Scholar 

  17. Roberts, A. W. and Varberg, D. E., Convex Functions, Academic Press, New York, London, 1973.

    Google Scholar 

  18. Spivak, M., A Comprehensive Introduction to Differential GeometryI-V, Publish or Perish, Inc. Berkeley, 1979.

    Google Scholar 

  19. Voss, A., Zur Theorie der Transformation quadratischer Differentialausdrücke und der Krümmung höherer Mannigfaltigkeiten, Mathematische Annalen 16(1880) 129–179.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rapcsák, T. An Unsolved Problem of Fenchel. Journal of Global Optimization 11, 207–217 (1997). https://doi.org/10.1023/A:1008245313867

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008245313867

Navigation