7 Lattice Points and Lattice Polytopes

Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation theory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the theory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 8.0.1.