Gruber P. Convex and Discrete Geometry

33.1 Fejes Tóth’s Inequality for Sums of Moments
33 Optimum Quantization 481
The 2-dimensional case of the problem has attracted the interest of the Hungarian
school of discrete geometry, ever since Fejes Tóth published his inequality on sums
of moments. By now, there are more than a dozen proofs of it known. The inequality
has applications to packing and covering problems for circular discs, to volume
approximation of convex bodies in E 3 by circumscribed convex polytopes and to
problems in areas outside of mathematics such as economics and geography. For a
planar generalization of it, to which Fejes Tóth, Fejes Tóth, Imre and Florian contributed,
see the report of Florian [337]. The inequality on sums of moments indicates
that, for certain geometric and analytic problems, regular hexagonal configurations
are close to optimal, and are possibly optimal.
In this section the sum theorem of Fejes Tóth will be presented.
For more information, see the book of Fejes Tóth [329] and the surveys of Florian
[337] and Gruber [438, 442].
Sums of Moments
Let f :[0, +∞) →[0, +∞) be non-decreasing, where f (0) = 0, and let H be a
convex 3, 4, 5, or 6-gon in E2 . Then, given a dissection C1,...,Cn of H and a set
S ={s1,...,sn} of n points in H or in Ed ,thesum
�
�
f (�x − si�)dx
i
Ci
is called a sum of moments. If f (t) = t 2 , this is a sum of moments of inertia. If for
the sets Ci we take the Dirichlet–Voronoĭ cells
Di = � x ∈ H :�x − si� ≤�x − s j� for j = 1,...,n � , i = 1,...,n,
in H corresponding to S, then the sum of moments decreases and, in addition,
�
�
�
f (�x − si�)dx = min{
f (�x − s�)}dx.
s∈S
i
Di
Fejes Tóth’s Inequality for Sums of Moments
Our aim is to prove the following result of Fejes Tóth [329].
Theorem 33.1. Let f :[0, +∞) →[0, +∞) be non-decreasing where f (0) = 0
and let H ⊆ E2 be a convex 3, 4, 5, or6-gon. Then,
(1) I (H, f, n) = inf
S⊆E2 �
�
#S=n
min{
f (�x − s�)}dx ≥ n
s∈S
f (�x�)dx,
H
H
Hn