Gruber P. Convex and Discrete Geometry

484 Geometry of Numbers
Now, combine (1), (2), (4), apply Jensen’s inequality 2.1 for convex functions,
and use (6) and (7) to see that
n�
�
a1 +···+an
I (H, f, n) ≥ M(ai,vi) ≥ nM
n
i=1�
|H|
�
≥ nM , 6 .
n
, v1
�
+···+vn
n
The proof of the theorem is complete. ⊓⊔
33.2 Zador’s Asymptotic Formula for Minimum Distortion
In higher dimensions, the problem raised in the introduction of Sect. 33 appeared first
in the work of Zador [1035] in the context of data transmission, more precisely, in
the problem to evaluate the quality of certain encoders as the number of code-words
of the code-books used tends to infinity. Later appearances are in error estimates for
numerical integration, approximation of probability measures by discrete probability
measures, approximation of convex bodies by polytopes and approximation of
functions by step functions. Far-reaching refinements and generalizations of Zador’s
results are due to Chernaya [206] and Gruber [443].
In the following, we give an asymptotic formula due to Zador for general d,
which is sufficient for many applications. For applications see Sect. 33.4.
Minimum Distortion
Let J be a Jordan measurable set in Ed with V (J) >0 and let α>0. The expression
�
I (J,α,n) = inf min
S⊆J s∈S
#S=n
{�x − s�α } dx
J
is called minimum distortion. This notion first appeared in the context of data transmission,
compare Sect. 33.4. In some cases the following slightly different quantity
is of interest:
K (J,α,n) = inf
S⊆Ed �
min
s∈S
#S=n
{�x − s�α } dx.
Zador’s Asymptotic Formula
J
The following result goes back to Zador [1035]. The proof is a simplified version of
the author’s [443] proof of a more general result. A body is a compact set which is
equal to the closure of its interior.
Theorem 33.2. Let α>0. Then there is a constant δ = δα,d > 0, depending only
on α and d, such that, for any Jordan measurable body J ⊆ E d with V (J) >0,
I (J,α,n), K (J,α,n) ∼ δ
V (J)α+d d
n α d
as n →∞.