A Continuum of Minimal Pairs of Compact Convex Sets which are not Connected by Translations

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].