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.