Gruber P. Convex and Discrete Geometry

29 Packing of Balls and Positive Quadratic Forms 435
Later we will see that perfect forms – besides being of interest per se – play an
important role for the determination of extreme forms or, more geometrically, for
the determination of lattice packings of B d of locally maximum density. Minkowski
[742] and Voronoĭ [1012] both proposed methods for finding all perfect forms in
d variables. Unfortunately, these methods are not efficient and it is still a task to
determine all perfect forms even for moderately large d.
If the minimum vectors of a positive definite quadratic form on E d do not
span E d , the form cannot be perfect. As a consequence of the above theorem and
Theorem 21.1 we thus obtain the following result:
Theorem 29.7. There are only finitely many non-equivalent perfect positive definite
quadratic forms on E d or, in other words, only finitely many equivalence classes of
perfect positive definite quadratic forms on E d , with the same minimum.
The numbers of non-equivalent perfect positive definite quadratic forms with
given minimum have been evaluated for dimensions d = 2,...,8:
d = 2: 1 : Lagrange [628]
d = 3: 1 : Gauss [364]
d = 4: 2 : Korkin and Zolotarev [611]
d = 5: 3 : Korkin and Zolotarev [611]
d = 6: 7 : Barnes [71]
d = 7: 33 : Jaquet-Chiffelle [542]
d = 8: 10916 : Dutour Sikiric and Schürmann [280]
For more information see Martinet [691] and Nebe [767].
Ryshkov’s Approach to Results of Korkin–Zolotarev and Voronoĭ on Perfect
and Eutactic Forms
In the following we describe an elegant path to the fundamental theorems of Korkin–
Zolotarev and Voronoĭ, where as a tool for the determination of extreme forms a generalized
convex polyhedron R(m) ⊆ P, m > 0, is used, dual to the so-called Voronoĭ
polyhedron. (For the definition of generalized convex polyhedra, see Sect. 14.2.)
It was introduced by Ryshkov [865], and we call it the Ryshkov’s polyhedron:
R(m) = � �
(a11,...,a1d, a22,...,add) ∈ E 1 2 d(d+1) : �
�
aik uiuk ≥ m .
u∈Z d \{o}
primitive
R(m) is the intersection of closed halfspaces in E 1 2 d(d+1) with (interior) normal
vectors (u2 1 , 2u1u2,...,2u1ud, u2 2 , 2u2u3,...,u2 d ).
of R(m):
We list several properties
Proposition 29.1. R(m) is closed, convex and has non-empty interior. Each ray in P
starting at the origin meets bd R(m) in precisely one point and, from that point on it,
is contained in int R(m).
i,k