ABSTRACT

This chapter lays the groundwork for the applications of lattice algebra that are addressed in Chapter 7. As the title indicates, the focus of this chapter is on extreme points of data sets. The chapter consists of two sections, with each having two subsections. Section 6.1 provides the reader with relevant concepts from convex set theory by including such notions as convex hulls, extreme points of convex sets, polytopes, and lattice polytopes. The novelty here is the notion of lattice polytopes of finite data sets. More specifically, for a given data set X, the polytope is simply defined as the intersection of the smallest interval containing X and the £-span of X.

The main focus of Section 6.2 is on the selection of a maximal number of affine independent extreme points of a lattice polytope. The assumption is that the lattice polytope is the smallest lattice polytope containing a finite n- dimensional data set X. The final result of this focus is a set of four sequential algorithms that select a maximal set of affine independent extreme points for n = 2, n = 3, n = 4 and n ≥ 4.