Abstract

Pairs of compact convex sets naturally arise in quasidifferential calculus as the sub- and superdifferentials of the directional derivative of a quasidifferentiable function (see [1]). Since the sub- and superdifferential in a given point are not uniquely determined, minimal representations are of special importance. For the 2-dimensional case, equivalent minimal pairs of compact convex sets are uniquely determined up to translations (see [2],[13]). For the 3-dimensional case, J. Grzybowski [2] gave an example of finitely many equivalent minimal pairs of compact convex sets which are not connected by translations. In this paper we construct for the 3-dimensional case a continuum of equivalent minimal pairs of compact convex sets which are not connected by translation for different indices. Moreover, we present a more general method of reducing pairs of compact convex sets by hyperplanes as in [7].

Author information

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

Diethard Pallaschke

Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Kaiserstr. 12, 76128 Karlsruhe, Germany.

Ryszard Urbański

Wydział Matematyki i Informatyki Uniwersytetu im Adama Mickiewicza, ul. Matejki 48/49, 60-769 Poznań, Poland

Keywords

  • Convex Analysis
  • quasidifferential calculus
  • pairs of compact convex sets

References

  1. V. F. Demyanov, A.M. Rubinov: Quasidifferential Calculus, Optimization Software Inc., Publication Division, New York, 1986.https://doi.org/10.1007/BFb0121132Google Scholar
  2. J. Grzybowski: On minimal pairs of convex compact sets, Archiv der Mathematik 63 (1994) 173–181.https://doi.org/10.1007/BF01189892Google Scholar
  3. J.-B. Hiriart-Urruty: Miscellanies on Nonsmooth Analysis and Optimization, in: Nondifferentiable Optimization: Motivations and Applications, Proceedings, Sopron, Hungary (1984) Eds.: V.F.Demyanov, D. Pallaschke, Lecture Notes in Economics and Mathem. Systems, Vol. 255, Springer, Heidelberg (1985) 8–24.https://doi.org/10.1007/978-3-662-12603-5_2Google Scholar
  4. L. Hörmander: Sur la fonction d’appui des ensembles convexes dans une espace localement convexe, Arkiv för Mat. 3 (1954) 181–186.https://doi.org/10.1007/BF02589354Google Scholar
  5. D. Pallaschke, S. Scholtes and R. Urbański: On Minimal Pairs of Compact Convex Sets, Bull. Acad. Polon. Sci., Sér. Sci. Math. 39 (1) (1991) 1–5.https://doi.org/10.68381/jca04001Google Scholar
  6. D. Pallaschke, R. Urbański: Some Criteria for the Minimality of Pairs of Compact Convex Sets, Zeitschrift für Operations Research (ZOR), Series Theory 37 (1993) 129–150.https://doi.org/10.1007/BF01414153Google Scholar
  7. D. Pallaschke, R. Urbański: Reduction of Quasidifferentials and Minimal Representations, Math. Programming, Series A 66 (1994) 161–180.https://doi.org/10.1007/BF01581143Google Scholar
  8. A. G. Pinsker: The space of convex sets of a locally convex space, Trudy Leningrad Engineering-Economic Institute, 63 (1966) 13–17.Google Scholar
  9. H. Rådström: An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165–169.https://doi.org/10.1090/S0002-9939-1952-0045938-2Google Scholar
  10. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton, New Jersey 1970.Google Scholar
  11. A. M. Rubinov, I. S. Akhundov: Differences of compact sets in the sense of Demyanov and its Application to Non-Smooth Analysis, Optimization 23 (1992) 179–189.https://doi.org/10.1080/02331939208843757Google Scholar
  12. G. T. Sallee: On the indecomposibility of the cone, Journ. London Math. Soc. 9 (1974) 363–367.https://doi.org/10.1112/jlms/s2-9.2.363Google Scholar
  13. S. Scholtes: Minimal pairs of convex bodies in two dimensions, Mathematika 39 (1992) 267–273.https://doi.org/10.1112/S002557930001500XGoogle Scholar
  14. R. Urbański: A generalization of the Minkowski-Rådström-Hörmander Theorem, Bull. Acad. Polon. Sci., Sér. Sci. Math. 24 (1976) 709–715.Google Scholar

Suggested citation

D. Pallaschke, R. Urbański. “A Continuum of Minimal Pairs of Compact Convex Sets which are not Connected by Translations.” Journal of Convex Analysis 3 (1996), No. 1, 83–95. https://doi.org/10.68381/jca03007

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1996. Rights now held by Banach Press.