Embedding theorems for classes of convex sets

Abstract

R/idstrOm'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 R~dstr6m's embedding theorem are proven which provide additional information on the embedding space. These results include those of H6rmander who proved a similar embedding theorem for the nonempty closed bounded convex subsets ofa Hausdorfflocally convex vector space. In contrast to HOrmander's approach via support function als, all embedding theorems of the present paper are proven by a refinement of Rgtdstr6m'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.

Embedding Theorems for Classes of Convex Sets in a Hypernormed Vector Space

“…•The previous results improve Proposition 4.1.In the case 3P = IR , Theorems 4.5 and 4.6 can be improved further and we obtain the following result[14; Theorem 7.8] which, at the same time, is a proper improvement of Rädström's embedding theorem:Extensions of Proposition 4.7 to the case of a Hausdorff locally convex vector space and to the class of all nonempty, closed, bounded, convex sets may be found in[14] and in this case m is said to be the midpoint of A , p is said to be the radius of A , and the set A is denoted by <m,p> . This midpoint-radius representation of a hypernorm ball, however, may fail to be unique, as pointed out in[2].…”

“…Rädström's embedding theorem and its generalizations also establish the preservation of topological properties. Furthermore, Kaucher [5] proved abstract embedding theorems which ensure the preservation of order properties, and embedding theorems for classes of convex sets in a Hausdorff locally convex vector Space which simultaneously guarantee the preservation of topological and order properties were proven by Hörmander [4] and Schmidt [14]. In view of these remarks, it is a natural question to ask whether or not Fischer's results can be improved.…”

Equivariant absolute extensor property on hyperspaces of convex sets

“…The main purpose of this section is to reconstruct the Rådström-Schmidt Embedding Theorem ( [14] and [17]) in order to prove that the hyperspaces cc(L) (and, in some cases, CB(L)) can be embedded as an invariant closed convex subset of a Banach G-space.…”

“…for every t ≥ 0 and A, B ∈ K. Details about this construction can be consulted in [14] and [17]. Now, suppose that L is a Banach G-space.…”

“…In this case, we can define a natural action on H(K) by the rule Proof. To see that H(K) is a Banach space, j is an isometry and j(K) is closed, the reader can consult [17] and [14]. We only prove here the facts concerning the action of G. Namely, we will prove that the action is well defined, continuous, isometric and that the embedding j is equivariant.…”