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

δ(L) =

vol(λ(L)Bd)

det L

=

λ(L)d vol Bd

det L

.

The PQF Q (and with it any arithmetically equivalent PQF obtained from some

basis of L) satisfies

(1.14) δ(L) = δ(Q) =

λ(Q)

(det Q)1/d

d/2

·

vol

Bd

2d

.

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 [180] 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

Q)1/d

·

4

(vol Bd)2/d

.

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

2d

πe

. 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(

√

rQA−1Bd)

= rQ

d/2

(det

Q)−1/2

vol

Bd

=

2d.

Minkowski discovered that not only ellipsoids of volume

2d

contain a non-zero

integral point, but also all other centrally symmetric convex bodies (non-empty,

compact convex sets).