1.3. THE SPHERE COVERING PROBLEM 11

Figure 6. Portion of the sphere covering given by the hexagonal lattice.

is equal to all of

Rd.

The covering radius µ(Λ) of Λ is defined by

µ(Λ) = inf{µ 0 : Λ +

µBd

=

Rd}.

The covering density of Λ is defined via the upper point density by

Θ(Λ) =

µ(Λ)d

vol

Bd

denssup Λ.

Note, in case of a lattice or a periodic set Λ, we can without problems replace the

upper point density by the point density dens Λ.

The density of the thinnest lattice sphere covering in

Rd

is denoted by

Θd = inf{Θ(L) : L ⊂

Rd

lattice },

and the density of the thinnest sphere covering by

Θd

∗

= inf{Θ(Λ) : Λ ⊂

Rd

a discrete set }.

As in the case of sphere packings, there exist thinnest lattice coverings and thinnest

coverings (cf. [137], [120]) attaining Θd and Θd.

∗

For such sets, in particular the

point density dens Λ is well defined.

As for the sphere packing problem, it has been conjectured by Zassenhaus (cf.

[121]), that Θd

∗

is attained by periodic sets. As in the case of sphere packings, it is

a big open problem to determine a dimension d with Θd Θd.

∗

In other words, it

is open whether or not there exists a dimension d with non-lattice coverings thinner

than any lattice covering. In particular, if Θ3

∗

= Θ3 holds is open (the “covering

analog of the Kepler conjecture”).

1.3.2. Known results — a snapshot. The first who considered the sphere

covering problem was Kershner, who showed in [151] that the hexagonal lattice

(see Figure 6) gives the best (thinnest) sphere covering in the planar case, even

without the restriction to lattices.

Since then the lattice covering problem has been solved up to dimension 5 (see

Table 2). In all these cases the lattice Ad

∗

(dual lattice of the root lattice Ad) with

Θ(Ad)

∗

=

d(d + 2)

12(d + 1)

d

(d + 1) ·

vol(Bd),

gives the optimal lattice covering. Gameckii [112], [113], and Bleicher [42] were

the first to compute the covering density of Ad

∗

for general d. They also showed that

Ad

∗

is locally optimal (to be made precise in Section 5.1.1) with respect to lattice

covering density in every dimension.