Gruber P. Convex and Discrete Geometry

If
22 Minkowski’s First Fundamental Theorem 371
V (Eϱ) = ϱ d 2 V (B d )
det(aik) 1 .
2
�
det(aik)
ϱ = 4
V (B d ) 2
� 1
d
,
then V (Eϱ) = 2 d . Now apply the fundamental theorem to the convex body Eϱ and
the lattice Z d . ⊓⊔
For smaller upper estimates, but in the terminology of density of packings
of balls, see Sect. 29.2 and the articles of Blichfeldt, Sidel’nikov, Levenstein and
Kabat’janskiĭ cited there. The best known upper bound is that of Levenstein and
Kabat’janskiĭ.
Minkowski’s Linear Form Theorem
This result can be formulated as follows, see Minkowski [735]:
Corollary 22.2. Let l1,...,ld be d real linear forms in d real variables, such that the
absolute value δ of their determinant is positive. Assume further that τ1,...,τd > 0
are such that τ1 ···τd ≥ δ. Then the following system of inequalities has a non-trivial
integer solution
|l1(x)| ≤τ1,...,|ld(x)| ≤τd.
Proof. Apply the fundamental theorem to the parallelotope P and the lattice Z d ,
where
P = � � 2
x :|l1(x)| ≤τ1,...,|ld(x)| ≤τd , V (P) = dτ1 ···τd
δ
≥ 2 d . ⊓⊔
The estimate in the linear form theorem cannot be improved for all systems of linear
forms.
Simultaneous Diophantine Approximation
As a consequence of the linear form theorem, we obtain a classical approximation
result due to Kronecker [618]. He proved it by means of Dirichlet’s pigeon hole
principle. Here, we follow Minkowski [735] who used his linear form theorem.
Corollary 22.3. Let ϑ1,...,ϑd ∈ R. Then the following system of inequalities has
infinitely many integer solutions (u0, u1,...,ud), where u0 �= 0:
�
�
�ϑ1 − u1
�
�
� ≤ 1
u0
u 1+ 1 d
0
�
�
,..., �ϑd − ud
�
�
� ≤ 1
u0
u 1+ 1 d
0
.