Gruber P. Convex and Discrete Geometry

430 Geometry of Numbers
The Layer Construction
A different construction for dense lattice packings of balls can be described as follows:
For d = 1 consider the packing {B 1 + u : u ∈ 2Z}. Assume now that d > 1
and that we have constructed a lattice packing of B d ,say{B d + l : l ∈ L d } where
L d is a lattice in E d .Ford + 1 proceed as follows: Consider E d as being embedded
in E d+1 as usual (first d coordinates). Clearly, {B d+1 + l : l ∈ L d } is a layer of
non-overlapping balls in E d+1 . Consider a translate {B d+1 + l + b : l ∈ L d } of this
layer where b = (b1,...,bd+1) is chosen such that bd+1 > 0 is minimal and such
that {B d+1 + l + ub : l ∈ L d , u ∈ Z} is a packing of B d+1 with packing lattice
L d+1 ={l + ub : l ∈ L d , u ∈ Z}.
This construction yields sequences of lattices (not necessarily unique, not necessarily
infinite). In this way one can obtain the densest lattice packing in dimensions
d = 2,...,8 and 24.
29.4 Geometry of Positive Definite Quadratic Forms and Ball Packing
The geometric theory of positive definite quadratic forms, which is related to lattice
packing and covering with balls, was developed since the nineteenth century
mainly by the Russian school of the geometry of numbers. Among the contributors
are Korkin, Zolotarev, Voronoĭ, Delone, Ryshkov and their students. Voronoĭ deserves
particular mention. We list also Minkowski, Blichfeldt, Watson and Barnes.
Recent contributions are due to the French school of quadratic forms of Martinet,
see [691].
Incidentally, note that the study of general, not necessarily positive definite
quadratic forms is a quite different thing. In the context of the geometry of numbers
it was cultivated, amongst others, by Bambah, Dumir, Hans-Gill, Raka and their
disciples at Chandigarh, see the report of Bambah, Dumir and Hans-Gill [57].
The main problem of the geometric theory of positive definite quadratic form is
to determine the extreme and the absolute extreme forms. This is equivalent to the
determination of the lattice packings of balls which have maximum density either
locally, i.e. among all sufficiently close lattice packings, or globally, i.e. among all
lattice packings. Other problems deal with minimum points, covering and reduction.
A quadratic form on E d may be represented by the vector of its coefficients in
E 1 2 d(d+1) . This allows us to transform certain problems on positive definite quadratic
forms and lattice packing of balls in E d into geometric problems about subsets of
E 1 2 d(d+1) which, in many cases, are more accessible. The solution of the geometric
problem, finally, is translated back into the language of positive forms. The systematic
use of this idea is due to Voronoĭ. A different application of it is our proof of
John’s characterization of the ellipsoid of maximum volume inscribed into a convex
body, see Theorem 11.2.
In the following we describe pertinent results of Korkin and Zolotarev, Voronoĭ,
Delone and Ryshkov, using the approach of Ryshkov [865].
In the book of Martinet [691], the theory of positive definite quadratic forms
is treated from an arithmetic point of view. A comprehensive modern geometric