Gruber P. Convex and Discrete Geometry

432 Geometry of Numbers
Lattice Packing of Balls and Positive Definite Quadratic Forms
First, let B be a non-singular d×d matrix, not necessarily symmetric. Then L = BZ d
is a lattice in E d . The columns b1,...,bd of B form a basis of L. ToL or, more
precisely, to the basis B we associate the positive definite quadratic form q on E d
defined by:
q(x) =�Bx� 2 = Bx · Bx = x T B T Bx = x T Ax for x ∈ E d ,
where A = (aik) = (bi ·bk) = B T B is a symmetric d ×d matrix. q is called the metric
form of L associated to the basis {b1,...,bd}. If{c1,...,cd} is a different basis
of L and C the matrix with columns c1,...,cd, then there is an integer unimodular
d × d matrix U such that C = BU. The metric form r of L associated to the basis
{c1,...,cn} then is
r(x) = x T C T Cx = x T U T B T BUx = x T U T AU x = q(Ux) for x ∈ E d
and thus is equivalent to q. If, on the other hand, r is a positive definite quadratic
form equivalent to q, then there is a basis of L such that r is the metric form of L
associated to this basis.
Second, let q(x) = x T Ax be a positive definite quadratic form on E d . Then by
a result from linear algebra, there is a d × d matrix B such that A = B T B. Thus
q is the metric form of the lattice L = BZ d associated to the basis {b1,...,bd}
which consists of the columns of B. Besides B it is precisely the matrices of the
form C = SB, where S is any orthogonal d × d matrix, for which A = C T C. Thus
L is unique up to (proper and improper) rotations.
Given a lattice L, we have, for the packing radius of B d with respect to L,
ϱ(B d , L) = 1
2 min � �l� :l ∈ L \{o} � = 1
2 λ1(B d , L).
The density of the lattice packing � ϱ(B d , L)B d + l : l ∈ L � is given by:
V � ϱ(B d , L)B d�
d(L)
= ϱ(Bd , L) d V (B d )
.
d(L)
We say that L provides a locally densest lattice packing of balls if
ϱ(B d , M) d V (B d )
d(M)
≤ ϱ(Bd , L) d V (B d )
d(L)
for all lattices M in a suitable neighbourhood of L.Ifq is a positive definite quadratic
form on E d , its arithmetic minimum min{q} is defined by:
min{q} =min � q(u) : u ∈ Z d \{o} � .
The points u ∈ Z d \{o}, for which min{q} is attained are the minimum points or
vectors of q. We say that q is an extreme form if