Representation of semigroups as systems of compact convex sets

Abstract

Under Minkowski addition and scalar multiplication the system of all compact convex subsets of R n {R^n} is an R-semigroup, i.e. a semigroup over the operator domain R of real numbers with certain conditions for the operation of R on the semigroup. Conversely, there is the question: When is an abstract R-semigroup isomorphic to a system S \mathfrak {S} of compact convex subsets of a suitable locally convex space? In this paper a necessary and sufficient condition for the existence of such a representation is given. This condition remains valid if, for the representing structures S \mathfrak {S} , systems of closed, bounded convex subsets with the closed Minkowski addition as addition are permitted. Finally, every R-semigroup of compact convex subsets of any locally convex space is isomorphic to a system of rectangular parallelepipeds of some vector space.