A Continuum of Minimal Pairs of Compact

Pairs of compact convex sets naturally arise in quasidiierential calculus as the sub-and superdiierentials of the directional derivative of a quasidiierentiable function (see 1]). Since the sub-and superdiierential 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 nitely 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 diierent indices. Moreover, we present a more general method of reducing pairs of compact convex sets by hyperplanes as in 7].