Abstract convexity spaces

Abstract

The principal question discussed in this dissertation is the problem of characterizing the linear and convex functions on generalized line spaces. A linear function is shown to be a convex function. The linear and convex functions are characterized, that is, a function f:[right arrow] X— is linear [convex] if and only if f[subscript l] is linear [convex] in the usual sense on each line of a generalized line space X. We prove that if a function has at least one support at each point on its graph, then it is a convex function. In the first chapter the basic concepts of abstract convexity spaces are introduced. The next chapter is concerned with join systems which are shown to be examples of abstract convexity spaces. On the other hand, a domain-finite, join-hull commutative abstract convexity space with regular straight segments satisfies the axioms of a join system. Consequently, such abstract convexity spaces satisfy the separation property. In Chapter III, the linearization of abstract spaces is done using a linearization family. The following chapter is on generalized line spaces and graphically it is shown that Pasch’s and Peano's axioms do not hold in a certain generalized line space. It is also proved that the separation property may not hold, in general, in a generalized line space. Finally, the convex and linear functions are studied on generalized line spaces. The linearization of generalized line spaces is done by means of the properties of a linearization family.