Pairs of compact convex sets arise in the quasidifferential calculus of
V.F. Demyanov and A.M. Rubinov as sub- and superdifferentials of
quasidifferen- tiable functions (see [26]) and in the formulas for the
numerical evaluation of the Aumann-Integral which were recently
introduced in a series of papers by R. Baier and F. Lempio (see [4],
[5], [10] and [9]) and R. Baier and E.M. Farkhi [6], [7],
[8]. In the field of combinatorial convexity G. Ewald et al. [36]
used an interesting construction called virtual polytope, which can also
be represented as a pair of polytopes for the calculation of the
combinatorial Picard group of a fan. Since in all mentioned cases the
pairs of compact con- vex sets are not uniquely determined, minimal
representations are of special to the existence of minimal pairs of
compact importance. A problem related convex sets is the existence of
reduced pairs of convex bodies, which has been studied by Chr. Bauer
(see [14]).