Gruber P. Convex and Discrete Geometry

370 Geometry of Numbers
it, is Dirichlet’s pigeon hole principle or, put analytically, the following simple
observation: If, on a measure space with total measure 1, the integral of a function is
greater than 1, then the function assumes values greater than 1.
Effective Methods for Finding Lattice Points
The fundamental theorem guarantees the existence of lattice points different from o
in C, but it does not tell us how to find such a point. In the context of algorithmic
geometry of numbers, a polynomial time algorithm has been specified which finds
such points, supposing that the volume of C is substantially larger than 2 d d(L).
See the remarks on the shortest lattice vector problem in Sect. 28.2 and, for more
information, Grötschel, Lovász and Schrijver [409].
22.2 Diophantine Approximation and Discriminants of Polynomials
The first fundamental theorem has numerous classical applications. These include
Minkowski’s applications to positive definite quadratic forms, Diophantine approximation,
his linear form theorem and discriminants of polynomials.
In the following, we present a selection of applications due to Minkowski and a
result of Lagrange on the representation of integers as sums of squares. For further
applications of the fundamental theorem. See, e.g. Gruber and Lekkerkerker [447]
and Schmidt [896].
Homogeneous Minimum of a Positive Definite Quadratic Form
Using a geometric argument on lattices and balls, Minkowski [732] was able to improve
a theorem of Hermite [495] on positive definite quadratic forms. At the time
when he published his result, Minkowski did not yet have the fundamental theorem
in its general form, but the argument used led him to the fundamental theorem shortly
afterwards. Minkowski’s result is as follows:
Corollary 22.1. Let
q(x) =
d�
aik xi xk for x = (x1,...,xd) ∈ E d
i,k=1
be a positive definite quadratic form (aik = aki). Then the following inequality has
a non-trivial integer solution u, i.e. u ∈ Z d \{o}:
�
det(aik)
q(u) ≤ 4
V (B d ) 2
� 1
d
.
Proof. To see this estimate, note that for ϱ>0thesetEϱ ={x : q(x) ≤ ϱ} is a
solid ellipsoid in E d with centre o and volume