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.
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