8 1. FROM QUADRATIC FORMS TO SPHERE PACKINGS AND COVERINGS
d lattice δd Hd author(s)
2 A2 0.9069 . . .
(
4
3
)1/2
Lagrange, 1773, [162]
3 A3 = D3 0.7404 . . .
21/3
Gauss, 1840, [114]
4 D4 0.6168 . . . 41/4 Korkin & Zolotarev 1877, [155]
5 D5 0.4652 . . .
81/5
Korkin & Zolotarev 1877, [155]
6 E6 0.3729 . . .
(
64
3
)1/6
Blichfeldt, 1934, [43]
7 E7 0.2953 . . .
641/7
Blichfeldt, 1934, [43]
8 E8 0.2536 . . . 2 Blichfeldt, 1934, [43]
24 Λ24 0.0019 . . . 4 Cohn & Kumar, 2004, [63]
Table 1. Known optimal lattice packings.
Theorem 1.3 (Minkowski’s Convex Body Theorem). Any centrally symmetric
convex body in Rd of volume 2d contains a non-zero integral point.
The theorem is sometimes also called “Minkowski’s first (fundamental) theo-
rem”. By its background story, the theory of PQFs and lattice sphere packings
may be regarded as the root of Minkowski’s Geometry of Numbers (cf. [183]).
1.2.4. Known results. The lattice sphere packing interpretation shows that
in order to obtain Hermite’s constant for a given dimension d, we may equivalently
try to determine the density of the densest lattice sphere packing
δd = sup
L⊂Rd
lattice
δ(L) = sup
Q∈Sd
0
δ(Q) = Hd
d/2
vol
Bd
2d
.
We show in Section 3.1 that this supremum is actually a maximum, which is attained
by at most finitely many densest lattice sphere packings in a given dimension. By
now the lattice sphere packing problem is mainly studied for its own beauty and
because of its connections to other fields such as coding and group theory (see
[69]). Table 1 lists the dimensions in which a solution to the lattice sphere packing
problem is known, together with the corresponding densities and values of the
Hermite constant.
The lattices Ad for d 2, Dd for d 3 and Ed for d = 6, 7, 8 are the so-
called root lattices. One of the most fascinating objects is the Leech Lattice Λ24 in
24 dimensions. Definitions and plenty of further information on these fascinating
lattices can be found for example in [69] (see also [188]). Some information on E8
and Λ24 is also in Section 5.5. Up to isometries, the lattices in Table 1 give unique
optimal solutions to the lattice sphere packing problem. In the 6-dimensional case,
this was first shown by Barnes [25]. In dimension 7 and 8 uniqueness has been
proved by Vetchinkin [261]. His paper also contains an extended treatment of
Blichfeldt’s work (cf. [266]). Optimality and uniqueness of the Leech lattice was
recently shown by Cohn and Kumar [63]. Their methods also provide a second,
completely different uniqueness and optimality proof for the lattice E8. A third
proof can be given by application of Voronoi’s algorithm. This approach is explained
in Section 3.1.
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