C-Minimal Pairs of Compact Convex Sets

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.