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]).
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