348 VIII The Geometry of Numbers
limn→∞γn/n≥ 1 / 2 πe= 0. 0585 ...,and it is possible that actually limn→∞γn/n= 1 / 2 πe. The significance of Hermite’s
constant derives from its connection with lattice packings of balls, as we now explain.
LetΛbe a lattice inRnandKa subset ofRnwhich is the closure of a nonempty
open setG. We say thatΛgives alattice packingforKif the family of translates
K+x(x∈Λ)is a packing ofRn, i.e. if for any two distinct pointsx,y∈Λthe inte-
riorsG+xandG+yare disjoint. This is the same as saying thatΛdoes not contain
the difference of any two distinct points of the interior ofK,sinceg+x=g′+yif
and only ifg′−g=x−y.IfKis a compact symmetric convex set with nonempty
interiorG, it is the same as saying that the interior of the set 2Kcontains no nonzero
point ofΛ, since in this caseg,g′∈Gimplies(g′−g)/ 2 ∈Gand 2g=g−(−g).
Thedensityof the lattice packing, i.e. the fraction of the total space which is
occupied by translates ofK, is clearlyλ(K)/d(Λ). Hence the maximum density of
any lattice packing forKis
δ(K)=λ(K)/∆( 2 K)= 2 −nλ(K)/∆(K),where∆(K)is the critical determinant ofK,asdefinedin§3. The use of the word
‘maximum’ is justified, since it will be shown in§6 that the infimum involved in the
definition of critical determinant is attained.
Our interest is in the special case of a closed ball:K=Bρ={x∈Rn:‖x‖≤ρ}.
By what we have said,Λgives a lattice packing forBρif and only if the interior of
B 2 ρcontains no nonzero point ofΛ, i.e. if and only ifm(Λ)^1 /^2 ≥ 2 ρ. Hence
δ(Bρ)=sup{λ(Bρ)/d(Λ):m(Λ)^1 /^2 = 2 ρ}
=κnρnsup{d(Λ)−^1 :m(Λ)^1 /^2 = 2 ρ},whereκn=πn/^2 /(n/ 2 )! again denotes the volume of the unit ball inRn. By virtue of
homogeneity it follows that
δn:=δ(Bρ)= 2 −nκnsup
Λγ(Λ)n/^2 ,where the supremum is now over all latticesΛ⊆Rn; that is, in terms of Hermite’s
constantγn,
δn= 2 −nκnγnn/^2.Thusγn, likeδn, measures the densest lattice packing of balls. A latticeΛ⊆Rnfor
whichγ(Λ)=γn, i.e. a critical lattice for a ball, will be called simply adensest lattice.
The densest lattice inRnis known for eachn≤8, and is uniquely determined apart
from isometries and scalar multiples. In fact these densest lattices are all examples of
indecomposable root lattices. These terms will now be defined.
A latticeΛis said to bedecomposableif there exist additive subgroupsΛ 1 ,Λ 2
ofΛ, each containing a nonzero vector, such that(x 1 ,x 2 )=0forallx 1 ∈Λ 1 and
x 2 ∈Λ 2 , and every vector inΛis the sum of a vector inΛ 1 and a vector inΛ 2 .Since
Λ 1 andΛ 2 are necessarily discrete, they are lattices in the wide sense (i.e. they are not