Gruber P. Convex and Discrete Geometry

The Selection Theorem of Mahler [680]
25 Mahler’s Selection Theorem 393
Theorem 25.1. Let (Ln) be a sequence of lattices in E d such that, for suitable constants
α, ϱ > 0, the following hold for n = 1, 2,...:
(i) d(Ln) ≤ α.
(ii) Ln is admissible for ϱB d .
Then the sequence of lattices contains a convergent subsequence.
We first present a familiar proof of Mahler’s theorem and then outline a beautiful
geometric proof due to Groemer [401], based on the notion of Dirichlet–Voronoĭ
cells and Blaschke’s selection theorem.
Proof. In the first step, the following will be shown:
(1) Let l1,...,ld be d linearly independent points of a lattice in E d . Then there
is a basis {b1,...,bd} of this lattice such that:
�bi� ≤�l1�+···+�li� for i = 1,...,d.
By Theorem 21.3, there is a basis {c1,...,cd} such that:
l1 = u11c1
l2 = u21c1 + u22c2
..........................
ld = ud1c1 +······+uddcd
where uik ∈ Z, uii �= 0.
Then
c1 = u −1
11 l1,
c2 = u −1
22 l2 + t21l1
..................................
cd = u −1
dd ld + tdd−1ld−1 +···+td1l1
where tik ∈ R.
Since {c1,...,cd} is a basis of L,thedvectors b1 = c1
= c1
b2 = c2 −⌊t21⌋l1
= c2 + v21c1
..................................................................
bd = cd −⌊td1⌋l1 −···−⌊tdd−1⌋ld−1 = cd + vdd−1cd−1 +···+vd1c1
where vik ∈ Z
also form a basis and
�bi� =�u −1
ii li + (tii−1 −⌊tii−1⌋)li−1 +···+(ti1 −⌊ti1⌋)l1�
≤|u −1
ii |�li�+�li−1�+···+�l1� ≤�li�+�li−1�+···+�l1�
for i = 1,...,d,
concluding the proof of (1).
The second step is to show the following, where α, ϱ are as in (i) and (ii):