Gruber P. Convex and Discrete Geometry

Next, we prove that
(18) lim inf
n→∞ I (J,α,n) n α d > δ V (J)α+d d
∼
λ1+ α .
d
Clearly, the following hold:
33 Optimum Quantization 489
(19) There are a subsequence of 1, 2,...,and constants σi > 0, i = 1,...,k,
such that:
I (J,α,n) n α d → lim inf
n→∞ I (J,α,n) n α d ,
n(Ci) ∼ σi �
n for i = 1,...,k,
σi ≤ 1
as n →∞in this subsequence.
i
Here we have applied (15) and (17). Finally, taking into account the fact that the
cubes Ci, i = 1,...,k, are pairwise disjoint by (16), the definitions of Sn(Ci), I ,
(11), (15), (19), Jensen’s inequality (Theorem 1.9) applied to the convex function
t → t −α/d , t > 0, and (16) together show the following:
I (J,α,n) n α �
d = min{�x
− s�
s∈Sn
J
α } dx n α d
≥ �
�
min
s∈Sn(Ci ) {�x − s�α } dx n α d ≥ �
i
Ci
∼ δ �
i
j
V (Ci) α+d
d
i
n α d
n(Ci) α d
j
∼ δ �
= δ �
V (C j) � V (Ci)
�
σi
�
V (C j) V (Ci)
≥ δ � � � V (Ci)
V (C j) �
V (C j)
j
� � � α+d
d
= δ V (C j)
� �
j
i
j
i
σi
i
i
I � Ci,α,n(Ci ) � n α d
V (Ci) α+d
d σ − α d
i
� − α d
� α
σi
− d
V (Ci)
� − α d ≥ δ V (J) α+d
d
λ α+d
d
as n →∞in the subsequence from Proposition (19). The proof of (18) is complete.
Since λ>1 was arbitrary, (18) immediately yields (12).
In the last step of the proof it will be shown that
(20) I (J,α,n) < ∼ δV (J) α+d
d
1
n α d
as n →∞.
Before proving (20), note that the definition of I yields the following inequality:
(21) Let C be a cube. Then I (C ∩ J,α,l) ≤ I (C,α,l).
,