An embedding theorem for spaces of convex sets

Abstract

so that the set of all convex sets of L is a commutative semigroup under addition. If the situation had been such that it was not only a semigroup but also a group, and if furthermore the second distributive law for multiplication with scalars had been true, then the set of convex sets would have been a vector space. The second distributive law: (X1+X2)A =X1A +X2A is, however, true if X1 and )X2 have the same sign, in particular if both are positive. It is therefore natural to ask whether the additive semigroup can be embedded in a group and whether multiplication with scalars can be extended to this group in such a way that the resulting system becomes a vector space, and so that for positive scalars the new multiplication coincides with the original one on the semigroup. The object of this paper is to prove such an embedding theorem for the case in which L is a normed linear space. We shall, however, not embed the class of all convex sets but only certain subclasses such as the class of all compact convex sets. Further we shall make the obtained vector space into a normed linear space by extending the Hausdorff distance [3, p. 146]1 between two convex sets. This is all done by using the well known classical method of extending commutative semigroups, which is used for example when defining negative numbers. The use of this method is made possible by the fact that the law of cancellation holds in our semigroup.