Gruber P. Convex and Discrete Geometry

22 Minkowski’s First Fundamental Theorem 369
(7) c(m) = 1
�
ψ(x)e
d(L)
F
−2πi m·x dx = 1
�
�
1(x + l)e
d(L)
F
l∈L
−2πi m·x dx
= 1 �
�
1(x + l)e
d(L)
l∈L
F
−2πi m·x dx
= 1 �
�
1(y)e
d(L)
−2πi m·(y−l) dy = 1
�
1(y)e
d(L)
−2πi m·y dy,
F
l∈L
F+l
by (5). Here i = √ −1 and F is a fundamental parallelotope of L. Note that the sums
are all finite and thus integration and summation may be interchanged. A similar
calculation shows that:
�
(8) ψ(x) 2 �
� � � �
dx = ψ(x) dx = 1(x + l) dx
l∈L
F
l∈L
F
F
= �
�
1(x + l) dx = �
�
�
1(y) dy = 1(y) dy
l∈L
F+l
�
1
= V
2 C
�
,
where we have used (6), (5) and the definition of 1. Finally, (8), Parseval’s theorem
for Fourier series, (7) and the definition of 1 show that:
�
1
V
2 C
� �
= ψ(x) 2 dx = d(L) �
|c(m)| 2
i.e.
F
m∈L ∗
= d(L)|c(o)| 2 + d(L) �
= d(L)
1
�
1
V
d(L) 2 2 C
2 d d(L) = V (C) + 4d d(L) 2
V (C)
m∈L ∗ \{o}
�2 �
m∈L ∗ \{o}
|c(m)| 2
+ d(L) �
m∈L ∗ \{o}
E d
E d
|c(m)| 2 ,
|c(m)| 2 ≥ V (C). ⊓⊔
Proof (by means of lattice packing). The reader who is not familiar with lattice
packing of convex bodies and the notion of density may wish to consult Sect. 30.1
first.
As before, it is sufficient to show (3). Statement (4) says that { 1 2C + l : l ∈ L} is
a lattice packing. The density of a lattice packing is at most 1. Hence
V ( 1 2 C)
d(L) ≤ 1, or V (C) ≤ 2d d(L). ⊓⊔
The Background of the Fundamental Theorem
The principle underlying the fundamental theorem and some of its refinements or
extensions, such as Blichfeldt’s [130] theorem or more modern generalizations of