Computational Geometry of Positive Definite Quadratic Forms. Polyhedral Reduction Theories, Algorithms, and Applications
The aim of this book is to give an introduction to computational geometry of a positive definite quadratic subject. The book can be considered as self-contained, describing on one side classical aspects of the theory (as developed by Minkowski, Voronoi and Delone), while on the other side keeping in touch with modern applications (such as lattice sphere packing and coverings). In the book, new proofs of known results are given in many places. A very good and readable presentation is combined with explicitly formulated algorithms that allow computer assisted experimentations. The book starts with an introduction containing basics of classical theory (the first chapter) and Minkowski’s reduction theory (the second chapter). The third and forth chapters are devoted to Voronoi’s first and second reduction theory. The fifth chapter “Local analysis of coverings and applications” is devoted to sphere covering and sphere packing-covering problems. The book is written in a lucid style and can be recommended to everybody interested in the theory of quadratic forms and its applications in lattice sphere packing and coverings.