Abstract

In a recent paper R. Urbański [13] investigated the mimimality of pairs compact convex sets which satisfy additional conditions, namely the minimal convex pairs. In this paper we consider some different possibilities of decomposing a given compact convex set into smaller compact convex sets which are related by translations or by reflections. Combining our results with the characterization of minimality of convex pairs of compact convex sets given in [13] we prove in the second part of this paper that for the two-dimensional case the following statements: M: equivalent minimal pairs of compact convex sets are uniquely determined up to translations (see [3], [11]) CM: equivalent convex minimal pairs of compact convex sets are uniquely determined up to translations (see [13]) are equivalent.

Author information

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

Diethard Pallaschke

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

Ryszard Urbański

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland

Keywords

  • Pairs of convex sets
  • sublinear function
  • quasidifferential calculus

References

  1. R. Baier: Mengenwertige Integration und die diskrete Approximation erreichbarer Mengen (Set-valued Integration and the discrete approximation of the reacheable set), Dissertation, University of Bayreuth, Bayreuther Mathematische Schriften, Heft 50 (1995).Google Scholar
  2. V. F. Demyanov, A. M. Rubinov: Quasidifferential Calculus, Optimization Software Inc., Publications Division, New York, 1986.https://doi.org/10.1007/BFb0121132Google Scholar
  3. J. Grzybowski: Minimal pairs of compact convex sets, Archiv der Mathematik 63 (1994) 173-181.https://doi.org/10.1007/BF01189892Google Scholar
  4. L. Hörmander: Sur la fonction d' appui des ensembles convexes dans un espace localement convex, Arkiv för Matematik 3 (1954) 181-186.https://doi.org/10.1007/BF02589354Google Scholar
  5. D. Pallaschke, P. Recht, R. Urbański: On locally Lipschitz quasidifferentiable functions in Banach spaces, Optimization 17 (1986) 287-295.https://doi.org/10.1080/02331938608843131Google Scholar
  6. D. Pallaschke, S. Scholtes, R. Urbański: On minimal pairs of compact convex sets, Bull. Polish Acad. Sciences, Math. 39 (1991) 105-109.Google Scholar
  7. D. Pallaschke, R. Urbański: Some criteria for the minimality of pairs of compact convex sets, Methods and Models of Operations Research (ZOR) 37 (1993) 129-150.https://doi.org/10.1007/BF01414153Google Scholar
  8. D. Pallaschke, R. Urbański: Reduction of quasidifferentials and minimal representations, Mathem. Programming 66 (1994) 161-180.https://doi.org/10.1007/BF01581143Google Scholar
  9. 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) 83-95.https://doi.org/10.68381/jca03007Google Scholar
  10. A. G. Pinsker: The space of convex sets of a locally convex space, Trudy Leningrad Engineering-Economic Institute, 63 (1966) 13-17.Google Scholar
  11. R. Urbański: On minimal convex pairs of convex compact sets, Arch. Math. 67 (1996) 226-238.https://doi.org/10.1007/BF01195239Google Scholar

Suggested citation

D. Pallaschke, R. Urbański. “Decompositions of Compact Convex Sets.” Journal of Convex Analysis 4 (1997), No. 2, 333–342. https://doi.org/10.68381/jca04018

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1997. Rights now held by Banach Press.