Gruber P. Convex and Discrete Geometry

486 Geometry of Numbers
I � K,α,n � �
= min{�x
− sni�
s∈Sn
K
α } dx = �
�
�x − sni�
i
Dni
α dx
≥ �
�
i
ϱni Bd �x� α �
V (Dni)
dx where ϱni =
V (B d � 1
d
)
= dV(Bd ) �
ϱ
α + d
i
α+d
ni =
dV(Bd )
(α + d) V (B d ) α+d
�
V (Dni)
d
i
α+d
d
d
≥
(α + d)V (B d ) α �
1 � � α+d
d d
n V (Dni) =
d n
i
(α + d) V (B d ) α d
For the proof of the upper estimate in (1), note that we can cover the cube K with
n balls of radius ϱn, say, such that the total volume of these balls is bounded above
by a constant independent of n. (Forn = l d take as set of centres a square grid of
edge-length 1 l in K . For general n choose l such that ld ≤ n 0, depending only on α and d such that:
Clearly, (1) implies,
(5) δ := lim inf
n→∞
I � K,α,n � ∼ δ
� n α d I � K,α,n �� ∈ R + .
n α d
as n →∞.
We show that one may replace lim inf in (5) by lim. Let λ>1 and choose k such
that:
(6) I � K,α,k � < λδ
.
k α d
First consider n of the form n = kl d , l = 1, 2,...Clearly, I � K,α,k � is the volume
of the mountain landscape
1
n α d
.