Geometry of numbers. 2nd ed.

*(English)*Zbl 0611.10017
North-Holland Mathematical Library, Vol. 37. Amsterdam etc.: North- Holland. XV, 732 p.; $ 95.50; Dfl. 215.00 (1987).

[For a review of the first edition (1969) see Zbl 0198.380.]

This book gives a rather complete coverage of the geometry of numbers. It includes relations to other branches of mathematics, in particular convex geometry, packing and covering, diophantine approximation, analytic number theory, coding theory, and numerical analysis.

This second edition was prepared jointly by P. M. Gruber and the author of the first edition; and it appears well-timed containing all progress made in this field during the last fifteen years. This progress was not uniform: This can be seen from the different lengths of the supplements which the authors added to the various chapters. The idea of the authors was to retain the existing text with minor corrections and to add the mentioned supplementary sections on more recent developments. Although this idea has some little drawbacks, it has the definite advantage of showing clearly where recent progress has taken place and where interesting new results may be expected. Moreover, these new sections can essentially be read independently from the original text.

As can be seen from the excellent supplements and the extensive bibliography, much new material can be found on ball packing, homogeneous and inhomogeneous minima, lattice polytopes and lattice point inequalities, Markov spectrum and zeta-functions, further on new relations to other branches of mathematics as, e.g., coding theory. Many implicitly or explicitly stated open problems will surely contribute to further research.

This excellent book can be used as a reference book as well as an advanced introduction to geometry of numbers.

This book gives a rather complete coverage of the geometry of numbers. It includes relations to other branches of mathematics, in particular convex geometry, packing and covering, diophantine approximation, analytic number theory, coding theory, and numerical analysis.

This second edition was prepared jointly by P. M. Gruber and the author of the first edition; and it appears well-timed containing all progress made in this field during the last fifteen years. This progress was not uniform: This can be seen from the different lengths of the supplements which the authors added to the various chapters. The idea of the authors was to retain the existing text with minor corrections and to add the mentioned supplementary sections on more recent developments. Although this idea has some little drawbacks, it has the definite advantage of showing clearly where recent progress has taken place and where interesting new results may be expected. Moreover, these new sections can essentially be read independently from the original text.

As can be seen from the excellent supplements and the extensive bibliography, much new material can be found on ball packing, homogeneous and inhomogeneous minima, lattice polytopes and lattice point inequalities, Markov spectrum and zeta-functions, further on new relations to other branches of mathematics as, e.g., coding theory. Many implicitly or explicitly stated open problems will surely contribute to further research.

This excellent book can be used as a reference book as well as an advanced introduction to geometry of numbers.

Reviewer: J.M.Wills

##### MSC:

11Hxx | Geometry of numbers |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

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

11Jxx | Diophantine approximation, transcendental number theory |

52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |

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