Gruber P. Convex and Discrete Geometry

(7) � (x, y) : x ∈ K, 0 ≤ y ≤ min{�x
− s�
s∈Sk
α } � ⊆ E d+1
33 Optimum Quantization 487
over the cube K with k valleys. Here Sk is a minimizing configuration in K , consisting
of k points. Each of the l d affinities
x1 → x1 + a1
l
,...,xd → xd + ad
l
a1,...,ad ∈{0, 1,...,l − 1}
, y → y
,
lα maps the landscape (7) onto a small landscape over a small cube of edge-length
1 l , where the small cubes tile the unit cube K . Thus, these small landscapes put
together, form a landscape over K with at most kl d valleys. This landscape contains
the following landscape:
� (x, y) : x ∈ K, 0 ≤ y ≤ min � �x − s� α : s ∈ 1
l (Sk + a), ai ∈{0, 1 ...,l − 1 �� .
Since this landscape is among those over which we form the infimum in the definition
of I (K,α,n = kl d ) it follows that
(8) I � K,α,n = kl d� d 1
≤ l
ldl α I � K,α,k � ≤ λδ
(kld ) α d
by (6). Secondly, we consider general n. Choose l0 so large that
� �
l0 + 1
(9) ≤ λ.
Then,
l0
(10) I � K,α,n � ≤ λ2 δ
n α d
for n ≥ kl d 0 .
To see this, let n ≥ kl d 0 , and choose l ≥ l0 such that kl d ≤ n < k(l + 1) d . Then the
definition of I , (8) and (9) yield (10):
I � K,α,n � n α d ≤ I � K,α,kl d�� k(l + 1) d� α �
l + 1
d ≤ λδ
l
� α
≤ λ 2 δ.
Since λ>1 was arbitrary, (5) and (10) together yield (4).
By applying a suitable affine transformation, we see that (4) implies the following
asymptotic formula:
(11) Let C ⊆ E d be a cube. Then I (C,α,n) ∼ δ
The third step of the proof is to show that
(12) I (J,α,n) > V (J)α+d d
∼ δ
n α d
as n →∞.
α+d
V (C) d
n α d
as n →∞.