Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  683.10025
Autor:  Erdös, Paul; Gruber, P.M.; Hammer, J.
Title:  Lattice points. (In English)
Source:  Pitman Monographs and Surveys in Pure and Applied Mathematics, 39. Harlow: Longman Scientific & Technical; New York etc.: John Wiley & Sons. viii, 184 p. \sterling 30.00 (1989).
Review:  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.
Reviewer:  E.Schulte
Classif.:  * 11Hxx Geometry of numbers
                   11-01 Textbooks (number theory)
                   52-01 Textbooks (convex and discrete geometry)
                   05-01 Textbooks (combinatorics)
                   52B99 Polytopes and polyhedra
                   11H06 Lattices and convex bodies (number theoretic results)
                   05B40 Packing and covering (combinatorics)
                   52C07 Lattices and convex bodies in n dimensions
                   11H55 Quadratic forms
                   05B45 Tessellation and tiling problems
                   52C17 Packing and covering in n dimensions (discrete geometry)
                   11H31 Lattice packing and covering (number-theoretic results)
Keywords:  lattice points; geometry of numbers; dissection problems; lattice polytopes; packing; covering; tiling problems; quadratic forms; crystallography; visibility; integral geometry; applications to numerical integration

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