# 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.