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.
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