Gruber P. Convex and Discrete Geometry

354 Geometry of Numbers
The aim of this chapter is to present basic results from these two areas. We start
with regular, i.e. lattice results and topics from the geometry of numbers such as
the fundamental theorems of Minkowski, the Minkowski–Hlawka theorem, the geometric
theory of positive definite quadratic forms and reduction. Minkowski’s fundamental
theorem and the Minkowski–Hlawka theorem are opposite cornerstones
of the geometry of numbers, the first yielding a simple upper bound for the density
of lattice packings, the second a lower bound. To show the reader the flavour
of the geometry of numbers of the English school we discuss some classical particular
arithmetic–geometric problems. Computational and algorithmic aspects are
touched. Then irregular, i.e. non-lattice results of a systematic character of discrete
geometry are presented, dealing with packing, covering, tiling, optimum quantization
and Koebe’s representation theorem for planar graphs. Besides quantization and
Koebe’s theorem, a further result of an instrumental character in classical discrete
geometry is Euler’s polytope formula, see Sect. 15.1. For corresponding problems
the irregular case for obvious reasons is more difficult than the regular one. Thus it
is not surprising that lattice results are, in general, much farther reaching than corresponding
non-lattice results. In our presentation the emphasis is on the geometry. The
given applications and relations to other areas deal with Diophantine approximation,
polynomials, error correcting codes, data transmission, numerical integration, graphs
and the Riemann mapping theorem. Lattice polytopes and some of their applications
were treated in Sect. 19 in the chapter on convex polytopes, but would also fit well
into the present chapter.
The reader who wants to get more detailed information is referred to the books
and surveys of Fejes Tóth [329, 330], Rogers [851], Gruber [416], Conway and
Sloane [220], Pach and Agarwal [783], Erdös, Gruber and Hammer [307], Cassels
[195], Gruber and Lekkerkerker [447], Ryshkov and Baranovskiĭ [867], Kannan
[563], Grötschel, Lovász and Schrijver [409], Siegel [937], Gruber [430], Zong
[1048,1049], Lagarias [625], Olds, Lax and Davidoff [778], Coppel [223], Ryshkov,
Barykinskiĭ and Kucherinenko [868], Matouˇsek [695], Böröczky [155], Ryshkov
[866], Bombieri and Gubler [148], to the collected or selected works of Minkowski
[745], Voronoĭ [1014], Davenport [246] and Hlawka [516] and to the pertinent
articles in the Handbooks of Convex Geometry [475] and of Discrete and Computational
Geometry [476] and in Discrete and Computational Geometry [273]. A large
collection of research problems in discrete geometry is due to Brass, Moser and
Pach [164].
Finite packing and covering problems, Erdös type problems, arrangements and
matroids will not be considered in the following. For these, see the book of Pach and
Agarwal [783], the monograph of Böröczky [155] on finite packing and covering,
the book of Matouˇsek [695] and the monograph of Bokowski [137] on oriented
matroids. Similarly, we consider lattice points in large convex bodies in the sense
of the circle problem of Gauss only in passing and instead refer to Gruber and
Lekkerkerker [447], Sect. iii, and articles in the Proceedings on Fourier Analysis
and Convexity [343], together with the references cited there.