On Convex Class of Pairs of Convex Bodies

In this paper we introduce a quotient class of pairs of convex bodies in which every member have convex union. The space of pairs of convex bodies has been investigated in a number of papers [3], [8], [9], and [12]. This space has found an application in quasidifferential calculus (cf. [1], [5], [7], [10]). A quasidifferential is represented as a pair of convex bodies and it is essential to find the minimal representation of this pair. The notion of minimal pairs was introduced in [5] and investigated in [2], [6], [7] and [11]. Some criteria of minimality are given in [6]. In this paper we investigate pairs of convex bodies with convex union. We introduced a quotient class of pairs of convex compact sets in which every member has convex union. Moreover some criteria for the convex class are given. In this paper X = (X, τ) stands for a real locally convex vector space, and X∗ denotes the dual space of X . Denote by K(X) the family of all convex bodies in X, i.e., 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 K(X) satisfies the order cancellation law; i.e. for every A,B,C ∈ K(X) the inclusion A + B ⊂ B + C implies A ⊂ C (cf. [12]).Hence it follows that K(X) endowed with the Minkowski sum is a commutative semigroup satisfying the law of cancellation. Now let K2(X) = K(X)×K(X); the equivalence relation between pairs of convex bodies is given by: (A,B) ∼ (C,D) if and only if A+D = B+C. For A,B ∈ K(X) we will use the notation A∨B := conv (A∪B), where the operation ”conv” denotes the convex hull. If A,B,C ∈ K(X), and b ∈ X , then A∨B+C = (A∨B)+C and A+ b = A+ {b}. We have [a, b] = {a} ∨ {b}. Let f ∈ X∗, A ∈ K(X) and c ∈ R. We denote by pA(f) := maxx∈Af(x) the support function of the set A. Moreover, H f := {x ∈ X | f(x) = c} and HfA := {x ∈ A | f(x) = pA(f)}, where H f is the hyperplane generated by the functional f and the number c, and HfA is the face of A with respect to f. For the sum of the faces of two convex bodies A,B ⊂ X with respect to f ∈ X∗ the identity Hf (A + B) = Hf + HfB holds true. For A ⊂ X we denote by ∂A the boundary Ā \Ao of the set A, where Ā := cl A and A := intA. Received by the editors June 12, 1996. 1991 Mathematics Subject Classification. Primary 52A07, 90C30, 26A27.