7 Further Remarks 357For otherwise there would exist a sequenceΛkof lattices inF such that either
m(Λk)→0ord(Λk)→∞, and clearly this sequence could have no convergent
subsequence.
We now prove the fundamentalcompactness theoremof Mahler (1946), which says
that this necessary condition onFis also sufficient.
Proposition 22If{Λk}is a sequence of lattices inRnsuch that
m(Λk)≥ρ^2 ,d(Λk)≤σ for all k,whereρ,σare positive constants, then the sequence{Λk}certainly has a convergent
subsequence.
Proof LetVkdenote the Voronoi cell ofΛk. We show first that the ballBρ/ 2 ={x∈
Rn:‖x‖≤ρ/ 2 }is contained in every Voronoi cellVk. In fact if‖x‖≤ρ/2 then, for
every nonzeroy∈Λk,
‖x−y‖≥‖y‖−‖x‖≥ρ−ρ/ 2 =ρ/ 2 ≥‖x‖,and hencex∈Vk.
Letvkbe a point ofVkwhich is furthest from the origin. ThenVkcontains the
convex hullCkof the setvk∪Bρ/ 2. Since the volume ofVkis bounded above byσ,so
also is the volume ofCk. But this implies that the sequencevkis bounded. Thus there
existsR>0 such that the ballBRcontains every Voronoi cellVk.
By Blaschke’s selection principle, the sequence{Vk}has a subsequence{Vkv}
which converges in the Hausdorff metric to a compact convex setV.SinceBρ/ 2 ⊆V,
it follows from Proposition 20 thatΛkv→Λ,whereΛis a lattice with Voronoi cellV.
To illustrate the utility of Mahler’s compactness theorem, we now show that, as
stated in Section 3, any compact symmetric convex setKwith nonempty interior has
a critical lattice.
By the definition of the critical determinant∆(K), there exists a sequenceΛk
of lattices with no nonzero points in the interior ofKsuch that d(Λk)→∆(K)as
k→∞.SinceKcontains a ballBρwith radiusρ>0, we havem(Λk)≥ρ^2 for allk.
Hence, by Proposition 22, there is a subsequenceΛkvwhich converges to a latticeΛ
asv→∞. Since every point ofΛis a limit of points ofΛkv, no nonzero point ofΛ
lies in the interior ofK.Furthermore,
d(Λ)= lim
v→∞
d(Λkv)=∆(K),and henceΛis a critical lattice forK.
7 FurtherRemarks
The geometry of numbers is treated more extensively in Cassels [11], Erd ̋oset al.[22]
and Gruber and Lekkerkerker [27]. Minkowski’s own account is available in [42].