1.2. LATTICES AND THE SPHERE PACKING PROBLEM 7
Figure 5. Portions of the lattice sphere packings with centers
given by the hexagonal and the face-centered-cubic (fcc) lattice.
kissing number, because it is the number of spheres in (1.13) which touch (“kiss”)
a fixed sphere of the packing.
The (sphere packing) density δ(L) of a lattice L is defined as the portion of
space covered by spheres, hence
λ(L)d vol Bd
The PQF Q (and with it any arithmetically equivalent PQF obtained from some
basis of L) satisfies
(1.14) δ(L) = δ(Q) =
Note that δ is not only invariant with respect to isometries (of the lattice L) and
with respect to arithmetical equivalent forms (of the PQF Q), but also with respect
to scaling. Equation (1.14) shows in particular that the supremum of possible lattice
packing densities is, up to a constant factor, equal to a power of Hermite’s constant.
Minkowski noticed  that the trivial bound
(1.15) δ(Q) ≤ 1,
which is an immediate consequence of the sphere packing interpretation, tremen-
dously improves the upper bound of Hermite for the arithmetical minimum (Corol-
lary 1.2). In fact, (1.15) is equivalent to
(1.16) λ(Q) ≤ (det
showing that the exponential constant on the right in Corollary 1.2 can be replaced
by a constant which grows roughly linear with d, namely as
. This trivial, but
significant improvement strengthened the geometric viewpoint and lead Minkowski
to a powerful fundamental principle. The ellipsoid E(Q, rQ), with rQ being the
right hand side in (1.16), has volume
vol E(Q, rQ) = vol(
Minkowski discovered that not only ellipsoids of volume
contain a non-zero
integral point, but also all other centrally symmetric convex bodies (non-empty,
compact convex sets).