Number Theory: An Introduction to Mathematics

6 Mahler’s Compactness Theorem 353

ofK. We will show that this defines a metric, theHausdorff metric, on the space of all
compact subsets ofRn.
Evidently

0 ≤h(K,K′)=h(K′,K)<∞.

Moreover h(K,K′)=0 impliesK=K′.Forifx′∈K′,thereexistxk∈Ksuch that
xk→x′and hencex′∈K,sinceKis closed. ThusK′⊆K, and similarlyK⊆K′.
Finally we prove the triangle inequality

h(K,K′′)≤h(K,K′)+h(K′,K′′).

To simplify writing, putρ =h(K,K′)andρ′ = h(K′,K′′).Foranyε>0, if
x ∈ Kthere existx′∈ K′such that‖x−x′‖<ρ+εand thenx′′∈ K′′such
that‖x′−x′′‖<ρ′+ε. Hence

‖x−x′′‖<ρ+ρ′+ 2 ε.

Similarly, ifx′′∈K′′there existsx∈Kfor which the same inequality holds. Sinceε
can be arbitrarily small, this completes the proof.
The definition of Hausdorff distance can also be expressed in the form

h(K,K′)=inf{ρ≥0:K⊆K′+Bρ,K′⊆K+Bρ},

whereBρ={x∈Rn:‖x‖≤ρ}. A sequenceKjof compact subsets ofRnconverges
to a compact subsetKofRnif h(Kj,K)→0asj→∞.
It was shown by Hausdorff (1927) that any uniformly bounded sequence of com-
pact subsets ofRnhas a convergent subsequence. In particular, any uniformly bounded
sequence of compact convex subsets ofRnhas a subsequence which converges to
a compact convex set. This special case of Hausdorff’s result, which is all that we
will later require, had already been established by Blaschke (1916) and is known as
Blaschke’s selection principle.

Proposition 20Let{Λk}be a sequence of lattices inRnand let Vkbe the Voronoi
cell ofΛk. If there exists a compact convex set V with nonempty interior such that
Vk→V in the Hausdorff metric as k→∞, then V is the Voronoi cell of a latticeΛ
andΛk→Λas k→∞.

Proof Since every Voronoi cellVkis symmetric, so also is the limitV.SinceVhas
nonempty interior, it follows that the origin is itself an interior point ofV. Thus there
existsδ>0 such that the ballBδ={x∈Rn:‖x‖≤δ}is contained inV.
It follows thatBδ/ 2 ⊆Vkfor all largek. The quickest way to see this is to use
Radstr ̊ om’s cancellation law ̈ , which says that ifA,B,Care nonempty compact con-
vex subsets ofRnsuch thatA+C⊆B+C,thenA⊆B. In the present case we have

Bδ/ 2 +Bδ/ 2 ⊆Bδ⊆V⊆Vk+Bδ/ 2 fork≥k 0 ,

and henceBδ/ 2 ⊆Vkfork≥k 0. Since alsoVk⊆V+Bδ/ 2 for all largek, there exists
R>0suchthatVk⊆BRfor allk.