Overview

This overview is intended to guide through the topics and results of this book.

Definitions and explanations for used terminology can be found with help of the

index. Readers not so familiar with the treated topics may perhaps start by looking

into the introductory material in Chapter 1 and Sections 2.1, 3.1 and 4.1. Most of

the results in this book are described in the language of positive definite quadratic

forms because it is the basis for most computations. In the following summary we

nevertheless primarily use “lattice-terminology”, since it is more common nowadays.

References to the literature can be found in the corresponding context of the book.

Chapters 1 and 2: An introduction. The first two chapters are mainly

introductory and contain, in contrast to the remaining chapters, only a few new

results.

In the first chapter we briefly recall the history of the theory of positive definite

quadratic forms and its natural connection to the lattice sphere packing problem.

Along the way, we introduce to basic notions and results used in the following

chapters. In Sections 1.2, 1.3 and 1.4 we review known results of our main aimed at

applications: the sphere packing, the sphere covering and the (simultaneous) sphere

packing-covering problem. For all three problems there is a notorious open question:

Do there exist dimensions in which lattices do not give optimal configurations? This

problem is one of our main motivations to extend Voronoi’s (lattice) reduction

theories in Chapter 3 and Chapter 4.

The second chapter deals with several aspects of Minkowski’s reduction the-

ory, which in a sense is a prototype for other polyhedral reduction theories. In

Section 2.1, we start with a short discussion about reduction in general, and then

introduce Minkowski’s theory in Section 2.2. We in particular give a previously

unknown, non-redundant description of Minkowski’s polyhedral reduction domain

up to dimension 7. In Section 2.3 we describe relations to Minkowski’s successive

minima and state a challenging conjecture concerning an improvement of a classical

and central theorem in the Geometry of Numbers. In Section 2.4 we end the second

chapter with an application of Minkowski reduction to multidimensional continued

fraction expansions, used for simultaneous Diophantine approximations.

Chapter 3: Voronoi’s first reduction theory. In Section 3.1 we start

by introducing to the theory of perfect lattices, respectively to “Voronoi’s first

reduction theory”. Based on so-called Ryshkov polyhedra, we give complete proofs

for the theory and explain Voronoi’s algorithm. We provide some background on

computational tools (such as the shortest vector problem and isometry tests for

lattices), which were recently used to finish the classification of perfect lattices up

to dimension 8.

xi