Gruber P. Convex and Discrete Geometry

424 Geometry of Numbers
lattice packing. Together with the original lattice packing of B d , it forms a packing
of translates of B d with density 2δL(B d ). In this dimension we then have,
δL(B d ) ≤ 1
2 δT (B d ).
29.2 Density Bounds of Blichfeldt and Minkowski–Hlawka
While the problem of finding tight lower and upper bounds or asymptotic formulae
as d → ∞ for the maximum density of lattice and non-lattice packings of balls
remains unsolved, there are substantial pertinent results.
In this section we give Blichfeldt’s upper estimate for δT (B d ) and the lower
estimate for δL(B d ) which follows from the Minkowski–Hlawka theorem. Then
more precise upper estimates for δT (B d ) due to Sidel’nikov, Kabat’janski and
Levenstein and lower estimates for δL(B d ) of Schmidt and Ball are described.
Upper Estimate for δT (B d ); Blichfeldt’s Enlargement Method
The following result of Blichfeldt [131] was the first substantial improvement of the
trivial estimate δT (B d ) ≤ 1.
Theorem 29.2. δT (B d ) ≤
d + 2
2
2 − d 2 = 2 − d 2 +o(d) as d →∞.
Proof. Thefirststepistoshowtheinequality of Blichfeldt:
(1)
n�
�t j − tk�2 ≤ 2n n�
�s − t j�2 for all t1,...,tn, s ∈ Ed .
Clearly,
j,k=1
j=1
n�
(a j − ak) 2 = �
(a 2 j + a2 k − 2a jak) = �
(a 2 j + a2 � �
k ) − 2
j,k=1
j,k
≤ �
(a 2 j + a2 �
k ) = 2n a 2 j for all a1,...,an ∈ R.
j,k
j
j,k
j
�2 a j
Then,
�
�t j − tk� 2 = �
�(s − t j) − (s − tk)� 2 = � � �
(si − t ji) − (si − tki) �2 j,k
j,k
≤ � � �
(si − t ji) − (si − tki) � � �
2
≤ 2n (si − t ji) 2
i
j,k
= 2n � �
(si − t ji) 2 = 2n �
�s − t j� 2 ,
j
i
j
concluding the proof of Blichfeldt’s inequality (1).
In the second step, we prove the following estimate, where the function f is
defined by f (r) = max � 0, 1 − 1 2 r 2� for r ≥ 0.
j,k
i
i
j