Gruber P. Convex and Discrete Geometry

30 Packing of Convex Bodies 441
Next, given a convex body C and a discrete set T , consider the family {C + t :
t ∈ T } of translates of C by the vectors of T .Itsupper and lower density are
lim sup
τ→+∞
1
(2τ) d
�
V � (C + t) ∩ τ K � 1
, lim inf
τ→+∞ (2τ) d
�
V � (C + t) ∩ τ K � .
t∈T
If they coincide, their common value is called the density δ(C, T ) of the given family.
In other sources different definitions are used. For packings and coverings with
translates of a convex body, these amount to the same values for the upper and lower
densities. For more general families of convex bodies our definitions are still close to
the intuitive notion of density and avoid strange occurrences such as packings with
upper density +∞.
Roughly speaking, the density of a family {C + t : t ∈ T } of translates of C
by the vectors of T is the total volume of the bodies divided by the total volume of
E d . In the case where this family is a packing, the density may be considered as the
proportion of E d which is covered by the bodies of the packing, or as the probability
that a ‘random point’ of E d is contained in one of the bodies of the packing.
Given a convex body C, letδT (C) and δL(C) denote the supremum of the upper
densities of all packings of translates of C, and all lattice packings of C, respectively.
δT (C) and δL(C) are called the (maximum) translative packing density and
the (maximum) lattice packing density of C, respectively. Clearly,
0 0 such that:
(1) F ⊆ σ K .
For the proof of the proposition, it is sufficient to show that
(2) 2d (τ − σ) d
d(L)
≤ #(L ∩ τ K ) ≤ 2d (τ + σ) d
d(L)
t∈T
for τ>σ.
The parallelotopes {F + l : l ∈ L} are pairwise disjoint and cover E d . Thus the
parallelotopes in this family which intersect τ K ,infact,coverτ K . By (1) these
parallelotopes are all contained in (τ + σ)K . Thus if m is their number, we have
#(L ∩ τ K ) ≤ m = mV(F)
V (F) ≤ V � (τ + σ)K �
V (F)
= 2d (τ + σ) d
.
d(L)