Gruber P. Convex and Discrete Geometry

29 Packing of Balls and Positive Quadratic Forms 431
exposition has yet to be completed, although parts of it can be found in the literature,
for example in Delone [257] and Delone and Ryshkov [259]. See also [1015].
The Cone of Positive Definite Quadratic Forms
Let q be a real quadratic form on E d ,
where
q(x) = x T Ax =
d�
aik xi xk for x ∈ E d ,
i,k=1
A = (aik), aik = aki,
is the symmetric d ×d coefficient matrix of q.Thecoefficient vector of q is the vector
(a11, a12,...,a1d, a22, a23,...,add) ∈ E 1 2 d(d+1) .
The determinant det A is the discriminant of q. In the following we will not distinguish
between a quadratic form, its coefficient matrix and its coefficient vector.
A quadratic form q on E d is positive definite if q(x) >0 for all x ∈ E d \{o}.
Let P be the set of all (coefficient vectors of) positive definite quadratic forms
on E d .Ifq, r ∈ P, then λq + µr ∈ P for all λ, µ > 0. Thus P is a convex cone in
E 1 2 d(d+1) with apex at the origin. A result from linear algebra says that a quadratic
form on E d is positive definite if and only if all principal minors of its coefficient
matrix are positive. Thus, if q ∈ P, all quadratic forms on E d , the coefficients of
which are sufficiently close to that of q, are also positive definite and thus are in P.
This means that P is open. P is called the (open convex) cone of (coefficient vectors
of) positive definite quadratic forms on E d . P is not a polyhedral cone although it
shares several properties with polyhedral cones. For more information on the algebraic
and geometric properties of P see the articles of Bertraneu and Fichet [103]
and Gruber [444].
Let U be an integer d × d matrix. It yields the linear transformation
x → Ux
of E d onto itself. In turn, this linear transformation induces a transformation
q(x) = x T Ax → q(Ux) = x T U T AU x
of the space of all quadratic forms on Ed onto itself, or in terms of coefficient vectors,
a transformation
�
�
U : (aik) →
�
l,m
uliumkalm
of the space of coefficient vectors onto itself. U may be considered as a linear transformation
of E 1 2 d(d+1) onto itself which maps P onto itself.
Two forms q, r ∈ P are equivalent, if there is an integer unimodular d ×d matrix
U such that r(x) = q(Ux). Since UZ d = Z d , equivalent forms assume the same
values for integer values of the variables, that is, on Z d .