Skip to main content
SpringerLink
Log in

Reduced Pairs of Compact Convex Sets and Ordered Median Functions

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

Abstract

We prove that in finite dimensional spaces every ordered median function is the Minkowski dual of a reduced pair of polytopes. This implies a very general theorem on the representation of an ordered median function as a uniquely determined difference of two sublinear functions up to adding and subtracting one and the same arbitrary sublinear function.

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

Fig. 1

Similar content being viewed by others

References

  1. Nickel, S., Puerto, J.: Location Theory—A Unified Approach. Springer, Berlin (2005)

    MATH  Google Scholar 

  2. Demyanov, V.F., Rubinov, A.M.: Quasidifferential calculus. Optimization Software Inc., Publications Division, New York (1986)

  3. Hörmander, L.: Sur la fonction d’ appui des ensembles convexes dans un espace localement convexe. Arkiv för Matematik 3, 181–186 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  4. Pallaschke, D., Urbański, R.: Pairs of Compact Convex Sets—Fractional Arithmetic with Convex Sets. Kluwer academic publishers, Dordrecht (2002)

    Book  MATH  Google Scholar 

  5. Melzer, D.: On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions. Math. Program. Stud. 29, 118–134 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bauer, Ch.: Minimal and reduced pairs of convex bodies. Geom. Dedic. 62, 179–192 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Demyanov, V.F., Rubinov, A.M. (eds.): Quasidifferentiability and Related Topics. Kluwer Academic Publisher, Dortrecht (2001)

    Google Scholar 

  8. Rådström, H.: An embedding theorem for spaces of convex sets. Proc. Am. Math. Soc. 3, 165–169 (1952)

    Article  MathSciNet  Google Scholar 

  9. Urbański, R.: A generalization of the Minkowski-Rådström-Hörmander theorem. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 24, 709–715 (1976)

    MATH  Google Scholar 

  10. Pinsker, A.G.: The space of convex sets of a locally convex space. Tr. Leningr. Eng. Econ. Inst. 63, 13–17 (1966)

    MathSciNet  Google Scholar 

  11. Urbański, R.: On minimal convex pairs of convex compact sets. Arch. Math. 67, 226–238 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Second expanded edition. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  13. Demyanov, V.F., Caprari, E.: Minimality of convex sets equivalent on a cone. Vestnik St. Petersburg Univ. Math. 31, 28–33 (1998)

  14. Grzybowski, J.: Minimal pairs of compact convex sets. Arch. Math. 63, 173–181 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grzybowski, J., Urbański, R.: Minimal pairs of bounded closed convex sets. Studia Math. 126, 95–99 (1997)

    MathSciNet  MATH  Google Scholar 

  16. Grzybowski, J., Urbański, R.: On the number of minimal pairs of compact convex sets that are not translates of one another. Studia Math. 158, 59–63 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pallaschke, D., Urbańska, W., Urbański, R.: C-minimal pairs of compact convex sets. J. Convex Anal. 4, 1–25 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Pallaschke, D., Urbański, R.: A continuum of minimal pairs of compact convex sets which are not connected by translations. J. Convex Anal. 3, 83–95 (1996)

    MathSciNet  MATH  Google Scholar 

  19. Pallaschke, D., Urbański, R.: On the separation and order law of cancellation for bounded sets. Optimization 51, 487–496 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Scholtes, S.: Minimal pairs of convex bodies in two dimensions. Mathematika 39, 267–273 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grzybowski, J., Nickel, S., Pallaschke, D., Urbański, R.: Ordered median functions and symmetries. Optimization 60, 801–811 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kalcsics, J., Nickel, S., Puerto, J., Tamir, A.: Algorithmic results for ordered median problems. Op. Res. Lett. 30, 149–158 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Puerto, J., Fernández, F.R.: Geometrical properties of the symmetrical single facility location problem. J. Nonlinear Convex Anal. 1, 321–342 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Shepard, G.C.: Decomposable convex polyhedra. Mathematika 10, 89–95 (1963)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Adam Mickiewicz University, 61-614, Poznań, Poland

    Jerzy Grzybowski & Ryszard Urbański

  2. University of Karlsruhe – KIT, 76128, Karlsruhe, Germany

    Diethard Pallaschke

Authors
  1. Jerzy Grzybowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Diethard Pallaschke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ryszard Urbański
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diethard Pallaschke.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grzybowski, J., Pallaschke, D. & Urbański, R. Reduced Pairs of Compact Convex Sets and Ordered Median Functions. J Optim Theory Appl 171, 354–364 (2016). https://doi.org/10.1007/s10957-015-0860-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0860-3

Keywords

Mathematics Subject Classification

Search

Navigation