Gruber P. Convex and Discrete Geometry

388 Geometry of Numbers
Hlawka [514] described the idea of his proof of the Minkowski–Hlawka theorem
as follows:
This theorem requires one to pick out from the infinite set of lattices, lattices which
have a particular property. Since it is difficult to find such lattices by chance, it is
plausible to make the following comparison: consider the problem of catching fish
of given length from a pond. Making one haul, one may catch such a fish only by
chance. For this reason it makes sense to catch many fish, hoping that a fish of the
desired length is among them. In probability theory this is called a random sample.
For this sample one considers the mean value, in our case the mean value of the
length, to get information on the sample. This was the idea which I applied in 1942 ...
The two proofs given earlier – as all other known proofs, including Hlawka’s
original proof – are based on mean value arguments dealing with huge sets of lattices.
This fact prevents the effective construction of lattices as specified in the theorem. In
particular, this explains why, so far, there is no effective algorithm available to construct
lattice packings of o-symmetric convex bodies of density at least 2 −d , although
such packings exist by the Minkowski–Hlawka theorem, compare Theorem 30.4. For
balls, the situation is slightly better but by no means satisfactory. There are constructions
of rather dense lattice packings of balls using codes. In many cases the codes
can be given effectively. Unfortunately the codes used by Rush [862] to reach the
Minkowski–Hlawka bound 2 −d+o(d) are not of this type. See Sect. 29.3.
Refinements
There are several refinements of the Minkowski–Hlawka theorem, see Gruber and
Lekkerkerker [447]. The best known estimate is due to Schmidt [893]: there is an
absolute constant α > 0 such that, for each Borel set B in E d with V (B) ≤
d log √ 2 − α, there is a lattice in E d of determinant 1 which contains no point of
B except, possibly, o. At present, it is the belief of many people working in the
geometry of numbers that, in essence, the theorem of Minkowski–Hlawka cannot be
refined. In the past, this was not always so, but Edmund Hlawka [517] told the author
that he was always convinced that no essential refinement was possible.
24.2 Siegel’s Mean Value Theorem and the Variance Theorem
of Rogers–Schmidt
A natural question to ask in the geometry of numbers is the following: Given a property
which a lattice may or may not have, is the set of lattices which have this property,
large or small? Tools which sometimes help to give an answer are measure
and Baire categories. Measure has turned out to be a versatile tool which applies
to many such questions, ever since Siegel [936] defined and put to use a natural
measure on the space of all lattices of determinant 1. Later contributions are due to
Rogers, Macbeath and Schmidt. Because of great technical difficulties, in particular
in the work of Schmidt, the development seems to have reached a deadlock. Despite
this, we believe that many important measure results in the geometry of numbers