Chapter 14 - Convexity

Abstract

This chapter presents the convexity theory in classical (unoriented) geometry. Convexity theory provides an important example of the advantages of the two-sided approach. In affine geometry, a set of points is said to be convex if it contains every segment whose endpoints lie in the set. By contrast in two-sided geometry, the segment joining two points is well defined and unique, as long as they are not antipodal. Moreover, this segment can be defined solely in terms of join, and so is a purely projective concept. Therefore, in two-sided geometry there is a notion of convexity that is preserved by projective maps, and yet retains most of the properties of affine convexity. Thus, in two-sided geometry one can use the tools of projective geometry to the development of theorems and algorithms involving convexity. Convexity and strict convexity are preserved by isomorphisms among projective spaces, in particular by projective maps of Tv to itself. The convex subsets of a two-sided space are closed under arbitrary intersections. Strictly convex sets are closed under arbitrary intersections.