1.2. LATTICES AND THE SPHERE PACKING PROBLEM 9

Probably one of the most challenging problems in the theory of (lattice) sphere

packings is how the density function behaves asymptotically. It is known to satisfy

−1

1

d

log2 δd −0.5990 . . . .

The lower bound is due to Minkowski (and in a more general form it is known as

Minkowski-Hlawka bound, cf. the work of Ball [8] for the currently “qualitative best

known” result and the recent work of Torquato and Stillinger [253] for a conjectural

improvement). The upper bound is due to Kabatyansky and Levenshtein [147],

[170] (cf. the work of Rogers [205] and Cohn and Elkies [60], [59] for best upper

bounds for d ≤ 42).

1.2.5. Sphere packings in general. The lower and upper bounds mentioned

above all apply to sphere packings in general, that is, to unions of spheres without

the restriction that the centers of spheres form a lattice. Starting with the work of

Thue [251] (cf. [108]) general sphere packings have been studied extensively. A lot

of attention, even outside the mathematical community, has been in particular on

the 3-dimensional sphere packing problem, also known as Kepler’s conjecture. After

some controversy (see [141], [158], [247], [107]; cf. [223] for another interpretation

of Kepler’s original quote), its proof by Hales recently has been published (cf. [129]

and [130]).

In order to define the general sphere packing problem, we need a notion of

density. We consider discrete sets Λ of Rd, that is, sets which have only finitely

many points in any bounded region. Let C = {x ∈ Rd : |xi| ≤ 1/2} be the d-

dimensional cube of unit edge length. Then we define the upper and lower (point)

density of a discrete set Λ as

(1.17)

denssup Λ = lim sup

λ→∞

card(Λ ∩ λC)

vol λC

and densinf Λ = lim inf

λ→∞

card(Λ ∩ λC)

vol λC

.

If both limits coincide, we simply speak of the point density of Λ, denoted by

dens Λ. Note that these notions depend on C and that there exist discrete sets

with denssup Λ = densinf Λ.

The packing radius of a discrete set Λ is simply defined by

λ(Λ) =

1

2

inf

x,y∈Λ,x=y

x − y .

The set

Λ+λ(Λ)Bd

is the sphere packing defined by Λ. In order to have a definition

of density available for all discrete sets Λ, the sphere packing density is defined by

by

δ(Λ) =

λ(Λ)d

vol

Bd

densinf Λ.

The density of the densest sphere packing is

(1.18) δd

∗

= sup{δ(Λ) : Λ ⊂

Rd

a discrete set }.

It turns out that there exist densest sphere packings, hence discrete sets Λ with

δd

∗

= δ(Λ) (cf. [137], [120]). For such Λ, upper and lower point density coincide

(cf. also the discussion in [156]).