Gruber P. Convex and Discrete Geometry

394 Geometry of Numbers
(2) Let L be a lattice in E d such that d(L) ≤ α and which is admissible for
ϱB d . Then there is a basis {b1,...,bd} of L with
�bi� ≤β, where β =
d 2 d α
ϱ d−1 V (B d ) ,
d(L) ≥ γ, where γ = ϱd V (B d )
2d .
To see this, choose d linearly independent points l1,...,ld ∈ L such that:
li ∈ λi bd B d or �li� =λi,
where λi = λi(B d , L). The assumptions in (2), together with Minkowski’s theorem
on successive minima, yield the following:
(3) ϱ ≤�li� ≤
2dd(L) �l1�···�li−1��li+1�···�ld�V (B d ) ≤
2dα ϱd−1V (B d ) .
Proposition (1) now shows that there is a basis {b1,...,bd} of L such that:
�bi� ≤
d 2dα ϱd−1V (B d = β.
)
This proves the first assertion in (2). The second assertion also follows from (3),
since �l1�,...,�ld� ≥ϱ by the assumptions in (2). The proof of (2) is complete.
In the final step of our proof, note that, by the assumptions of the theorem and
(2), there are bases {bn1,...,bnd} of Ln for n = 1, 2,..., such that the following
hold:
ϱ ≤�bn1�,...,�bnd� ≤β and | det{bn1,...,bnd}| = d(Ln) ≥ γ.
Now apply a Bolzano–Weierstrass type argument to the first basis vectors, then to
the second basis vectors, etc., to get the selection theorem. ⊓⊔
Proof (outline, following Groemer). Let α, ϱ > 0 be as in the theorem. Consider, for
n = 1, 2,...,theDirichlet–Voronoĭ cell Dn = D(Ln, o) of o with respect to Ln,
Dn = � �
x :�x� ≤�x − l� for all l ∈ Ln .
It consists of all points of E d which are at least as close to o as to any other point
of Ln. Since Ln is admissible for ϱB d ,wehave,
(4) ϱ
2 Bd ⊆ Dn.
Since {Dn + l : l ∈ Ln} is a tiling of E d , i.e. both a packing and a covering, we
conclude that V (Dn) = d(Ln) ≤ α. Noting that Dn is convex, (4) then implies that
(5) Dn ⊆ β B d .