Gruber P. Convex and Discrete Geometry

496 Geometry of Numbers
Fig. 33.1. Densest lattice covering with circular discs in E 2
(7) Let {B 2 + s : s ∈ S} be a covering (of E 2 ). Then
1
4τ 2
�
s∈S
≥
≥
A � (B 2 + s) ∩ τ K �
4(τ − 2)2
4τ 2
4(τ − 2)2
4τ 2
1
4(τ − 2) 2
2π
√ 27 for τ>2
�
s∈S
(B 2 +s)∩(τ−2)K �=∅
A(B 2 + s)
by (6), applied to (τ − 2)K instead of τ K .
Proposition (7) shows that θT (B 2 ) ≥ 2π/ √ 27. Since there are lattice coverings
of B 2 of density 2π/ √ 27, the equality θT (B 2 ) = θL(B 2 ) = 2π/ √ 27 follows
(Fig. 33.1). ⊓⊔
Minimum Distortion of Vector Quantization
The scheme of a data transmission system is as follows:
x −→ encoder
C
−→ channel
C ′
−→ decoder −→ x ′ .
A source produces signals. The signals are the input of the encoder. The encoder
assigns to each incoming signal x a code-word c which is taken from a code-book
consisting of finitely many code-words. The code-word c then is transmitted in a
channel. The output of the channel is a word c ′ , possibly different from c, from
which the decoder produces a signal x ′ . We study the quality of the encoder in a
particularly important special case, compare Gray and Neuhoff [394].
The signals are the points of a Jordan measurable set C in E d such that V (C)>0.
The encoder consists of a dissection of C into n Jordan measurable sets, say
C1,...,Cn, thecells, and a set {c1,...,cn} of n points in E d ,thecode-book. The