Gruber P. Convex and Discrete Geometry

and therefore
�x − m� 2 = �
29 Packing of Balls and Positive Quadratic Forms 421
i
µ 2 i � ˆbi� 2 ≥ 1
4 � ˆbk� 2 +
≥ 1
4 � ˆbk� 2 +
d�
i=k+1
λ 2 i � ˆbi� 2
d�
i=k+1
�x − l�2 =� �
λi
i
ˆbi� 2 ≤ 1
k�
� ˆbi�
4
i=1
2 +
d�
≤ 1
k�
2
4
k−i � ˆbk� 2 +
d�
λ 2 i � ˆbi� 2
i=1
i=k+1
i=k+1
≤ 2 k−2 �x − m� 2 ≤ 2 d−2 �x − m� 2 ,
µ 2 i � ˆbi� 2
λ 2 i � ˆbi� 2
where we have used the inequality (4) in the proof of Theorem 28.1. The proof of (4)
and thus of the corollary is complete. ⊓⊔
29 Packing of Balls and Positive Quadratic Forms
Packing of balls is a story with many chapters starting with Kepler [576, 577]. The
corresponding theory of positive definite quadratic forms dates back to Lagrange
[628] and Gauss [364]. It seems that the main reason for the intensive geometric work
on lattice packing of balls in the nineteenth and twentieth century was the arithmetic
background. Rogers [851], p.2, expressed this in his little classic on Packing and
Covering as follows:
Largely because of its connection with the arithmetic minimum of a positive definite
quadratic form, much effort has been devoted to the study of δL(Kn), whereKn is
the unit sphere in n-dimensional space.
For the notions of packing, upper density, maximum density δL(B d ) of lattice
packings of B d and maximum density δT (B d ) of packings of translates of B d ,see
Sect. 30.1.
In this section we outline a selection of classical and modern results and methods.
We begin with densest lattice packing of balls in 2 and 3 dimensions and the
density bounds of Blichfeldt and Minkowski–Hlawka. Then a chapter on the geometry
of positive definite quadratic forms is presented, which goes back at least to
Korkin and Zolotarev and to Voronoĭ. Finally, relations between ball packing and
error-correcting codes are discussed.
For more information, compare the monographs of Thompson [995], Conway
and Sloane [220], Leppmeier [650], Zong [1049] and Martinet [691] and the surveys
of Bambah [56] and Pfender and Ziegler [799].