Abstract

Pairs of compact convex sets naturally arise in quasidifferential calculus as a sub- and superdifferentials of a quasidifferentiable function (see [1]). Since the sub- and superdifferential 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 [3], [14]). For the 3-dimensional case, this is not longer true. J. Grzybowski [3] gave an example of finitely many equivalent minimal pairs of compact convex sets which are not connected by translations. A continuous family of equivalent minimal pairs of compact convex sets which are not connected by translation for different indices is given in [9]. In a recent paper R. Urbański [16] investiged the mimimality of pairs of compact convex sets which satisfy additional conditions, namely the minimal convex pairs. This paper is a continuation of this research direction. Here we study the minimality under a different type of conditions. Moreover we give a definition of a “convex hull” of a pair of sets.

Author information

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D. Pallaschke

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

W. Urbańska

Poznań University of Technology, Department of Mathematics, Piotrowo 3a, PL-60965 Poznań, Poland.

R. Urbański

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

Keywords

  • Pairs of convex sets
  • sublinear function
  • quasidifferential calculus

References

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Suggested citation

D. Pallaschke, W. Urbańska, R. Urbański. “C-Minimal Pairs of Compact Convex Sets.” Journal of Convex Analysis 4 (1997), No. 1, 1–25. https://doi.org/10.68381/jca04001

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