On minimal convex pairs of convex compact sets

摘要

1. Introduction. The space of pairs of convex compact sets has been investigated in a number of papers (see [3], [7], [8], [11]). Recently this space has found application in quasidifferential calculus (see [1], [9]). A quasidifferential is represented as a pair of convex compact sets and it is essential to find a minimal representation of this pair. The notion of minimal pairs was introduced in [4]. Some criteria of minimality are given in [5], In this paper we investigate pairs of convex compact sets with convex union. We show for every pair of convex compact sets there exists an equivalent pair of sets with convex union. This observation allows us to introduce a new type minimality; convex minimality. In this paper X — {X, x) will be a Hausdorff topological vector space. Let Jf (X) denote the family of all nonempty compact convex subsets of X, If A, B are nonempty subsets of X then A + B is the usual algebraic Minkowski sum of A and B, It may be showed that Jf(X) satisfies the order cancellation law, i.e. for A, B, CeJf(X) the inclusion A+ BaB + C implies AcC (see [11]). From this it follows that Jf (X) together with Minkowski sum is a commutative semigroup satisfying the law of cancellation. Now let X2(X) = Jf [X)x Jf (X). The equivalence relation between pairs of convex compact sets is given as follows (A, B) (C,D)ifiA + D = B + C. The set M'2 (X) may be ordered by the relation: {A, B) g (C, D) iff A c C and Bel). For A, Betf(X) we will use the notation A v B = conv (A u B), A denotes the closure of A. If A, B, C e JT (X), and b e X, then (A v B) + C = A v B + C and A + {b} = A + b, The interval [a, b] = a v f;. A pair (A, B) e jf 2(X) is called minimal if for any pair (C, D) equivalent to (A, B) the relation (C, D) g (A, B) implies that (A, B) = (C, D). In [4] it has been proved that for any [A, B) e Jf2[X) there exists a minimal pair {Ao, Bo) equivalent to (A, B).