Skip to main content
SpringerLink
Log in

Fenchel Problem of Level Sets

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

An Erratum to this article was published on 02 April 2008

Abstract

The Fenchel problem of level sets in the smooth case is solved by deducing a new and nice geometric necessary and sufficient condition for the existence of a smooth convex function with the level sets of a given smooth pseudoconvex function. The main theorem is based on a general differential geometric tool, the space of paths defined on smooth manifolds. This approach provides a complete geometric characterization of a new subclass of pseudoconvex functions originating from analytical mechanics and a new view on convexlike and generalized convexlike mappings in image analysis.

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

Access this article

Log in via an institution

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. T. Rapcsák (2005) ArticleTitleLocal Convexity on Smooth Manifolds Journal of Optimization Theory and Applications 127 162–176

    Google Scholar 

  2. Rapcsák T. Survey of the Fenchel Problem of Level Sets, Variational Analysis and Applications, Edited by F. Giannessi and A. Maugeri. Kluwer Academic Publisher, Dordrecht, Holland (in print).

  3. O.L. Mangasarian (1965) ArticleTitlePseudo-Convex Functions SIAM Journal on Control 3 281–290 Occurrence Handle10.1137/0303020

    Article  Google Scholar 

  4. J.P. Crouzeix (1998) Characterizations of Generalizd Convexity and Generalized Monotonicity: A Survey J.P. Crouzeix J.E. Martinez-Legaz M. Volle (Eds) Generalized Convexity Generalized Monotonicity Kluwer Academic Publishers Dordrecht, Holland 237–256

    Google Scholar 

  5. W. Fenchel (1951) Convex Cones, Sets, and Functions, Mimeographed Lecture Notes Princeton University Press Princeton, New Jersey

    Google Scholar 

  6. M. Avriel W.E. Diewert S. Schaible I. Zang (1998) Generalized Concavity Plenum Press New York, NY

    Google Scholar 

  7. S. Schaible I. Zang (1980) ArticleTitleOn the Convexifiability of Pseudoconvex C2-Functions Mathematical Programming 19 289–299 Occurrence Handle10.1007/BF01581649

    Article  Google Scholar 

  8. W. Fenchel (1956) ArticleTitleÜber Konvexe Funktionen mit vorgeschriebenen Niveaumannigfaltigkeiten Mathematische Zeitschrift 63 496–506 Occurrence Handle10.1007/BF01187955

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer and Automation Research Institute, Hungarian Academy of Sciences, Budapest, Hungary

    T. Rapcsák

Authors
  1. T. Rapcsák
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This paper is dedicated to the memory of Guido Stampacchia.

The author thanks K. Balla for assistance in solving the system of differential equations of Example 2.1 and for helpful remarks.

This research was supported in part by the Hungarian Scientific Research Fund, Grants OTKA-T043276 and OTKA-T043241, and by CNR, Rome, Italy.

An erratum to this article is available at http://dx.doi.org/10.1007/s10957-008-9360-z.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rapcsák, T. Fenchel Problem of Level Sets. J Optim Theory Appl 127, 177–191 (2005). https://doi.org/10.1007/s10957-005-6399-y

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-005-6399-y

Keywords

Search

Navigation