minimal pairs of compact convex sets.
Diethard Pallaschke, W. Urbańska, Ryszard Urbański · 1997
Pairs of compact convex sets naturally arise in quasidierential calculus as a sub- and superdierentials of a quasidierentiable function (see [1]). Since the sub- and superdierential 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 nitely 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 dierent indices is given in [9]. In a recent paper R. Urbanski [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 dierent type of conditions. Moreover we give a denition of a \\convex hull " of a pair of sets.