Gruber P. Convex and Discrete Geometry

456 Geometry of Numbers
Theorem 31.1. Let C be a proper convex body in E d with o ∈ int C. Then there is a
covering lattice L of C such that δ(C, L) = ϑL(C).
Proof. Let (Ln) be a sequence of covering lattices of C such that:
(1) δ(C, L1) =
Then
V (C)
d(L1) ≥ δ(C, L2)
V (C)
=
d(L2) ≥···→ϑL(C) ≥ 1.
(2) V (C) ≥ d(Ln).
Let B be a solid Euclidean ball with centre o such that B ⊇ C. Since Ln is a covering
lattice of C and thus a fortiori of B, it follows that for the covering radius µ(B, Ln)
of B with respect to Ln that µ(B, Ln) ≤ 1. Thus Jarník’s transference theorem 23.4
implies that, for the dth successive minimum of B with respect to Ln, wehave
(3) λd(B, Ln) ≤ 2µ(B, Ln) ≤ 2.
Considering the inequality
λ1(B, Ln) ···λd(B, Ln)V (B) ≥ 2d
d ! d(Ln),
which holds by the theorem 23.1 on successive minima, it follows from (1) and
(3) that
say. Thus
λ1(B, Ln) ≥
2d d(Ln)
d ! λd(B, Ln) d−1V (B) ≥
2d V (C)
d ! 2d−1 = α,
V (B)δ(C, L1)
(4) Ln is admissible for the ball αB with centre o.
Noting (2) and (4), Mahler’s selection theorem 25.1 implies that the sequence
(Ln) has a convergent subsequence. After appropriate cancellation and renumbering
of indices, if necessary, we may assume that L1, L2, ···→ L, where L is a suitable
lattice. The definition of convergence of lattices implies the following propositions:
(5) d(L1), d(L2), ···→d(L).
(6) If ln ∈ Ln are such that l1, l2, ···→l ∈ E d , then l ∈ L.
For the proof that
(7) L is a covering lattice of C,
let x ∈ E d . We have to show that x ∈ C + l for suitable l ∈ L. Since, by assumption,
the lattices Ln are covering lattices of C, there are vectors ln ∈ Ln such that