Gruber P. Convex and Discrete Geometry

444 Geometry of Numbers
respectively. By the Brunn–Minkowski theorem 8.1 and the inequality of Rogers and
Shephard 9.10, these densities are related as follows:
δ(C, L) ≤ δ(D, L) ≤ 1
2 d
� �
2d
δ(C, L) ∼
d
2d
√ δ(C, L).
πd
There is equality in the left inequality if and only if C is centrally symmetric and in
the right inequality if and only if C is a simplex.
Admissible and Critical Lattices
Let C be a convex body in E d with o ∈ int C. A lattice L is admissible for C if o is
the only point of L in int C. L is critical for C if it is admissible and has minimum
determinant among all admissible lattices. This determinant is denoted ∆(C) and is
called the critical determinant of C. The lattice L is locally critical for C if it is
admissible and has minimum determinant among all admissible lattices in a suitable
neighbourhood of it.
Proposition 30.4. Let C be a proper convex body, D = 1 2 (C − C) its central symmetrization
and L a lattice in Ed . Then the following statements are equivalent:
(i) {C + l : l ∈ L} is a packing.
(ii) {D + l : l ∈ L} is a packing.
(iii) L is admissible for 2D.
Proof. Note Proposition 30.3 and its proof. Then
{C + l : l ∈ L} is a packing
⇔{D + l : l ∈ L} is a packing
⇔ int D ∩ (int D + l) =∅for each l ∈ L \{o},
⇔ l �∈ int D − int D for each l ∈ L \{o},
⇔ l �∈ int 2D for each l ∈ L \{o}
⇔ L is admissible for 2D. ⊓⊔
Corollary 30.2. Let C be a proper convex body, D = 1 2 (C − C) its central
symmetrization and L a lattice in Ed . Then the following statements are equivalent:
(i) {C + l : l ∈ L} is a packing of maximum density.
(ii) L is a critical lattice of 2D = C − C.
Note that
δL(C) =
V (C)
∆(2D) .