AN EMBEDDING THEOREM FOR UNBOUNDED CONVEX SETS IN A BANACH SPACE

Abstract

In this paper we embed the space of nonempty closed convex subsets of a Banach space into a Banach space. Namely, we prove the following result: Theorem. Let X be a Banach space, V a closed convex cone in X, dH a Hausdor distance. Let CV =fA X : A nonempty closed and convex, dH(A;V ) <1g: Then CV is a commutative semigroup with cancellation law and Hausdor metric dH is positively homogeneous and invariant under translation. Consequently CV can be embedded isometrically and isomorphically as a convex cone into a Banach space.