Gruber P. Convex and Discrete Geometry

Generalizations
33 Optimum Quantization 491
From the point of view of applications, it is of interest to extend Zador’s asymptotic
formula to asymptotic formulae for expressions of the following form, where f is a
non-decreasing function on [0, +∞), J ⊆ Ed Jordan measurable, �·�a norm on Ed ,
possibly different from the standard Euclidean norm, and w : J → R + a continuous
weight function:
�
(24) inf
S⊆J
#S=n
J
or of the form
�
(25) inf
S⊆J
#S=n
J
min{
f (�x − s�)}w(x) dx,
s∈S
min
s∈S { f (ρM(x, s))}w(x) dωM(x),
where 〈M,ρM,ωM〉 is a d-dimensional Riemannian manifold with Riemannian metric
ρM and measure ωM, J ⊆ M Jordan measurable and w : J → R + a weight
function as before. Similar problems arise if the sets S are not necessarily contained
in the set J. For many functions f corresponding asymptotic formulae exist. For
pertinent results and more information compare Gruber [442, 443].
The Constant δ
The proof of the following asymptotic formula was communicated by Karoly
Böröczky [157].
Proposition 33.1. Let α>0. Then δα,d ∼ d α 2
(2πe) α 2
as d →∞.
Proof. We make use of the material in the first two steps of the proof of the theorem
of Zador. Note that
� α
δα,d = lim n d I (K,α,n)
n→∞
� .
Estimate below: From the proof of (1) we have,
(26) n α d
d I (K,α,n) ≥
(α + d) V (B d ) α .
d
Estimate above: By Rogers’s covering theorem 31.4, for all sufficiently large
n there are points {xn1,...,xnn} in the cube K and ρn > 0 such that the balls
ρn B d + xni cover the cube and their total volume is less than 2d log d. Then
n ρ d n V (Bd ) ≤ 2d log d, or
n α d ρ α n ≤ (2d log d) α d
V (B d ) α d
.