Gruber P. Convex and Discrete Geometry

The First Fundamental Theorem
22 Minkowski’s First Fundamental Theorem 367
A very satisfying answer to the above problem is the following first fundamental
theorem of Minkowski [734] or Minkowski’s first theorem:
Theorem 22.1. Let C be an o-symmetric convex body and L a lattice in E d such that
V (C) ≥ 2 d d(L). Then C contains a pair of points ±l ∈ L \{o}.
The first proof is based on Dirichlet’s pigeon hole principle and a formula to
calculate the volume of a convex body. The idea to use the pigeon hole principle
in this context seems to be due to Scherrer [886]. The second proof is due to Siegel
[935] and makes use of Fourier series and Parseval’s theorem. The third one utilizes a
close relation between the fundamental theorem and the notion of density of a lattice
packing of convex bodies (Fig. 22.1).
Proof (using the pigeon hole principle). Since C is compact and L discrete, it is
sufficient to prove the theorem under the stronger assumption that
(1) V (C) >2 d d(L).
The convex body 1 2C is Jordan measurable by Theorem 7.4. Using the substitution
rule for multiple integrals, the asymptotic formula (3) in Sect. 7.2 to calculate the
Jordan measure then implies that
�
1
V
2 C
� �
1 1
∼ # C ∩
2 n L
� �
1
· d
n L
�
as n →∞.
Since by (1) V ( 1 2C)>d(L), it follows that
�
1 1
(2) # C ∩
2 n L
�
> n d for all sufficiently large n.
Keep such an n fixed and choose a basis {b1,...,bd} of L. To each point
u1
n b1 +···+ ud
n bd ∈ 1 1
C ∩
2 n L
Cd
o
Fig. 22.1. Fundamental theorem