Gruber P. Convex and Discrete Geometry

33 Optimum Quantization 485
The constant δ = δ2,d−1 has appeared already in the context of volume approximation
of convex bodies by circumscribed convex polytopes, see Theorem 11.4. The
formula
δα,d ∼ d α 2
(2πe) α 2
as d →∞
will be proved later, see Proposition 33.1.
We prove only the asymptotic formula for I (J,α,n). The proof for K (J,α,n) is
very similar, but in one detail slightly more complicated.
Proof. In the following const stands for a positive constant which may depend on α
and d. If const appears several times in the same context, this does not mean that it
is always the same constant.
In the first step of the proof we show that
(1) const
n α d
≤ I � K,α,n � ≤ const
n α , where K ={x : 0 ≤ xi ≤ 1}.
d
For the proof of the lower estimate some preparations are needed. First, the following
will be shown:
(2) Let I ⊆ E d be a convex body and let ϱ > 0 and s ∈ E d be such that
V (I ) = V (ϱB d + s). Then
�
I
�x − s� α dx ≥
�
ϱB d +s
�x − s� α dx =
�
ϱB d
�x� α dx.
To see this, note that the sets I \(ϱB d + s) and (ϱB d + s)\I have the same volume
and �x − s� α is greater on the first set. Second, dissecting ϱB d into infinitesimal
shells of the form (t + dt)B d \tB d of volume dV(B d )t d−1 dt and adding, we obtain,
(3)
�
ϱB d
�x� α dx = dV(B d �
)
0
ϱ
t d−1 t α dt = d
α + d V (Bd )ϱ α+d .
After these preparations, choose minimizing configurations Sn ={sn1,...,snn}
⊆ K for I � K,α,n � and consider the corresponding Dirichlet–Voronoĭ cells in K ,
Dni = � x ∈ K :�x − sni� ≤�x − snj� for j = 1,...,n � , i = 1,...,n.
The cells Dni are convex polytopes which tile K and are such that, for x ∈ Dni,
one has �x − sni� α ≤�x − snj� α for j = 1,...,n. This together with (2), (3) and
Jensen’s inequality, applied to the convex function t → t (α+d)/d , yields the lower
estimate in (1):