Gruber P. Convex and Discrete Geometry

488 Geometry of Numbers
Arguments similar to those which led to (1) yield the inequalities
(13) const
≤ I (J,α,n) ≤ const
.
n α d
n α d
Choose minimizing configurations Sn ={sn1,...,snn} ⊆J for I (J,α,n) and consider
the corresponding Dirichlet–Voronoĭ cells in J,
Dni = � x ∈ J :�x − sni� ≤�x − snj� for j = 1,...,n � , i = 1,...,n.
Since J is a body and I (J,α,n) → 0asn →∞by (13), a simple indirect compactness
argument implies that
(14) max{diam
Dni} →0asn →∞.
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Next, we introduce some necessary notation. If C is a cube, then
n(C) = #{i : C ∩ Dni �= ∅}, Sn(C) ={sni : C ∩ Dni �= ∅}.
Now n(C) will be estimated below and above:
(15) Let C ⊆ J be a cube. Then const n ≤ n(C) ≤ n.
Clearly,
I � C,α,n(C) � �
≤
C�
=
C
min
s∈Sn(C) {�x − s�α } dx
min{�x
− s�
s∈Sn
α } dx ≤ I (J,α,n) ≤ const
n α d
by (13). Hence I � C,α,n(C) � → 0 and thus n(C) →∞as n →∞. This, in turn,
shows by (11) that
I � C,α,n(C) � ∼ δ
α+d
V (C) d
n(C) α d
as n →∞.
Hence n(C) ≥ const n, concluding the proof of (15).
For the proof of (12) let λ>1. Since J is compact and Jordan measurable, the
following holds:
(16) There are pairwise disjoint cubes C1,...,Ck ⊆ J such that:
�
i
V (Ci) ≥ 1
V (J).
λ
Since the cubes C1,...,Ck are pairwise disjoint, it follows from (14) that the
sets Sn(Ci), i = 1,...,k, are also pairwise disjoint and thus
(17) �
n(Ci) ≤ n for sufficiently large n.
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