Acta Applicandae Mathematica

, Volume 5, Issue 3, pp 209-237

First online:

Embedding theorems for classes of convex sets

  • Klaus D. SchmidtAffiliated withSeminar für Statistik, Universität Mannheim, A5

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Rådström'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 Rådström's embedding theorem are proven which provide additional information on the embedding space. These results include those of Hörmander who proved a similar embedding theorem for the nonempty closed bounded convex subsets of a Hausdorff locally convex vector space. In contrast to Hörmander's approach via support functionals, all embedding theorems of the present paper are proven by a refinement of Rådström'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.

AMS (MOS) subject classifications (1980)

Primary: 52A07 46A40 secondary: 28B20 47H10 54C60 60D05 65G10 90A14

Key words

Classes of convex sets embedding theorems topological vector lattices integration of random sets limit theorems for random sets fixed-point theorems for set-valued maps mathematical economics interval mathematics