Gruber P. Convex and Discrete Geometry

24 The Minkowski–Hlawka Theorem 389
still await discovery. Category on the space of lattices has rarely been used, a minor
exception is the article of Aliev and the author [23].
In this section we describe the natural measure on the space of lattices of
determinant 1 and state an elegant generalization of the Minkowski–Hlawka theorem,
Siegel’s mean value theorem. Then, an important adjunct result is specified
which, perhaps, is best called the variance theorem of Rogers and Schmidt.
The reader who wants to get more information on measure theory on spaces of
lattices is referred to the books of Rogers [851], Gruber and Lekkerkerker [447]
and Siegel [937].
Measure on the Space of Lattices of Determinant 1
Let L(1) be the space of all lattices of determinant 1 in Ed . Since, by Mahler’s
selection theorem 25.1, respectively, its Corollary 25.1, this space is locally compact,
it carries many measures. A measure on this space, which has proved extremely
useful for the geometry of numbers, can be defined as follows.
Let SL(d) be the locally compact multiplicative group of all real d × d matrices
with determinant 1, the special linear group. We consider the Haar measure on
SL(d). Up to normalization, this measure can be described as follows: Representing
a d × d matrix as a point in Ed2 , the space SL(d) is a surface in Ed2 .Givena
Borel set B ⊆ SL(d), consider the cone with basis B and apex at the origin, that is
the set {λB : 0 ≤ λ ≤ 1, B ∈ B} ⊆Ed2 . This set is again Borel and its Lebesgue
measure is the Haar measure of B.
Let U be the sub-group of SL(d), consisting of all integer unimodular matrices.
Given a lattice L ∈ L(1), any basis of L with positive determinant can be identified
with a matrix B ∈ SL(d). The family of all bases of L, with positive determinant, is
then the set of all matrices of the form BU where U ∈ U, that is, a left coset of U.
There exists a fundamental domain F of SL(1) with respect to U, that is a set which
contains precisely one matrix from each coset of U, which is Borel and has positive
finite Haar measure. Clearly, the Haar measure can be normalized such that F has
measure 1. Since F contains precisely one basis of each lattice in L(1), this gives
a measure µ on the space of all lattices of determinant 1 in Ed . This measure was
defined by Siegel [937] using reduction theory of positive definite quadratic forms,
thus following an idea of Minkowski.
Siegel’s Mean Value Theorem
Using this measure, Siegel [936] proved the following elegant result.
Theorem 24.2. Let f : Ed → R be Riemann integrable. Then
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f (l) : l ∈ L \{o} dµ(L) = f (x) dx.
L(1)
E d