Embedding theorems for classes of convex sets

Abstract

Radstrom's embedding theorem states that the nonempty compact convex subsets of a normed vector space can be identified with points of another normed vector space such that the embedding map is additive, positively homogeneous, and isometric. In the present paper, extensions of Radstrom's embedding theorem are proven which provide additional information on the embedding space. These results include those of Hormander who proved a similar embedding theorem for the nonempty closed bounded convex subsets of a Hausdorff locally convex vector space. In contrast to Hormander's approach via support functionals, all embedding theorems of the present paper are proven by a refinement of Radstrom's original method which is constructive and does not rely on Zorn's lemma. This paper also includes a brief discussion of some actual or potential applications of embedding theorems for classes of convex sets in probability theory, mathematical economics, interval mathematics, and related areas.