Gruber P. Convex and Discrete Geometry

494 Geometry of Numbers
33.4 Packing and Covering of Circles, Data Transmission and Numerical
Integration
Fejes Tóth’s theorem on sums of moments and Zador’s theorem on minimum distortion,
the refinements of the latter by the author [443] and the results on the structure
of the minimizing configurations are tools for a series of applications, see the references
cited in the introduction of Sect. 33.
In the following, applications to packing and covering with circular discs, to
data transmission and to numerical integration are presented. For an application of
Zador’s theorem to the approximation of convex bodies by circumscribed convex
polytopes see the approximation theorem 11.4.
Packing and Covering of B 2
Using Fejes Tóth’s theorem on sums of moments, we prove a result of Thue [996,
997] and Fejes Tóth [327] which says that the maximum density of a packing in E 2
with circular discs equals the maximum lattice packing density, compare Sects. 29.1
and 31.4.
Corollary 33.1. δT (B 2 ) = δL(B 2 ) = π
√ 12 = 0.906 899 ...
Proof. We first show a weaker statement:
(1) Let τ>0 and consider a packing of n translates of B 2 in the square τ K ,
where K ={x :|xi| ≤1}. Then
τ K
nπ π
≤ √ .
4τ 2
12
Let Sn be the set of centres of this packing. Fejes Tóth’s inequality on sums of
moments then shows that
�
�
�
0for0≤ t ≤ 1,
(2) min{
f (�x−s�)} dx ≥ n f (�x�) dx where f (t) =
s∈Sn
1fort > 1,
Hn
and Hn is a regular hexagon of area A(Hn) = A(H)/n = 4τ 2 /n with centre o.
Note that
�
(3) min{
f (�x − s�)} dx = 4τ
s∈Sn
2 − nπ
and
τ K
�
(4) n
Hn
f (�x�) dx = nA(Hn\B 2 ) ≥ nA(Hn) − nA(B 2 ) = 4τ 2 − nπ.