CHAPTER 3.2 - Lattice Points

Abstract

This chapter focuses on lattice points and presents various lattice point problems, both from a theoretical and an algorithmic point of view. Lattice points are used in areas including numerical analysis, computer science, and, in particular, integer programming. Minkowski's theorem has numerous applications in the geometry of numbers. Its impact is in the number of ramifications, refinements, and generalizations that it led to. The classical theory of geometry of numbers deals mainly with centrally symmetric convex bodies; hence, there are only a few classical results on general convex bodies. One such result is the simple mean value theorem. Lattice point problems are closely related to lattice packing and lattice covering. There are various inequalities for lattice-point-free convex bodies that involve width. Results of such a kind are of some relevance for reducing a d-dimensional integer programming problem to lower dimensional ones. The methods in the theory of lattice polytopes are mainly from combinatorics and linear algebra, and several results do not require convexity.