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**Lattice points.**
*(English)*
Zbl 0683.10025

Pitman Monographs and Surveys in Pure and Applied Mathematics, 39. Harlow: Longman Scientific & Technical; New York etc.: John Wiley & Sons. viii, 184 p. £30.00 (1989).

This excellent book collects together many fascinating ramifications of the concept of lattice points.

Historically lattice point theory originated as a central subject in the geometry of numbers. Accordingly the authors give a quite detailed exposition of the classical and modern contributions of the geometry of numbers. Besides they touch on many other topics dealing with dissection problems; lattice polytopes; packing, covering, and tiling problems; quadratic forms; crystallography; visibility; connections with integral geometry; and applications to numerical integration, combinatorics, graph theory, and others.

This book is highly recommended to anyone interested or working in lattice point theory. It provides an exposition of the subject with only a few proofs but in general quite detailed (intuitive) explanations of the results. This way the book becomes (as the authors put it) an “appetizer” for further study. I am convinced that the book will considerably stimulate further research in the area of lattice points.

Historically lattice point theory originated as a central subject in the geometry of numbers. Accordingly the authors give a quite detailed exposition of the classical and modern contributions of the geometry of numbers. Besides they touch on many other topics dealing with dissection problems; lattice polytopes; packing, covering, and tiling problems; quadratic forms; crystallography; visibility; connections with integral geometry; and applications to numerical integration, combinatorics, graph theory, and others.

This book is highly recommended to anyone interested or working in lattice point theory. It provides an exposition of the subject with only a few proofs but in general quite detailed (intuitive) explanations of the results. This way the book becomes (as the authors put it) an “appetizer” for further study. I am convinced that the book will considerably stimulate further research in the area of lattice points.

Reviewer: E.Schulte

### MSC:

11Hxx | Geometry of numbers |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

52-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to convex and discrete geometry |

05-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics |

52B99 | Polytopes and polyhedra |

11H06 | Lattices and convex bodies (number-theoretic aspects) |

05B40 | Combinatorial aspects of packing and covering |

52C07 | Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry) |

11H55 | Quadratic forms (reduction theory, extreme forms, etc.) |

05B45 | Combinatorial aspects of tessellation and tiling problems |

52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |

11H31 | Lattice packing and covering (number-theoretic aspects) |