Gruber P. Convex and Discrete Geometry

436 Geometry of Numbers
Proof. As an intersection of closed halfspaces, R(m) is closed and convex. The
definition of R(m) implies that each ray R in the open convex cone P starting at
the origin intersects R(m) in a half-line. Let R be such a ray and consider the
first point of the halfline R ∩ R(m), sayA. Next choose 1 2d(d + 1) further such
rays, say Ri, i = 1,..., 1 2d(d + 1), inP which determine a simplicial cone which
contains the ray R in its interior. For any B ∈ R ∩ R(m), B �= A, wemay
choose points Ai ∈ Ri ∩ R(m) such that B is an interior point of the simplex
conv � �
A, A1,...,A 1
2 d(d+1) which, in turn, is contained in the convex set R(m).
This implies that int R(m) �= ∅and thus concludes the proof of the proposition. ⊓⊔
Proposition 29.2. R(m) ⊆ P.
Proof. Consider a point in R(m) and let q be the corresponding quadratic form.
We have to show that q is positive definite. By the definition of R(m) we have q(u) ≥
m > 0 for each u ∈ Z d \{o}. Thus q(r) >0 for each point r ∈ E d \{o}, with rational
coordinates and therefore q(x) ≥ 0 for each x ∈ E d by continuity. Thus q is positive
semi-definite. If q were not positive definite, then a well known arithmetic result says
that for any positive number and thus in particular for m, there is a point u ∈ Z d \{o},
with q(u) less than this number. This is the required contradiction. ⊓⊔
Proposition 29.3. The points of R(m), respectively, of bd R(m) correspond precisely
to the positive definite quadratic forms with arithmetic minimum at least m, respectively,
equal to m.
Proof. This is an immediate consequence of the definition of R(m) and the earlier
two propositions. ⊓⊔
Proposition 29.4. Let U be an integer unimodular d × d matrix and let U be the
corresponding transformation of P. Then U � R(m) � = R(m) and U � bd R(m) � =
bd R(m).
Proof. This follows from Proposition 29.3 by taking into account that with any positive
definite quadratic form having arithmetic minimum at least m or equal to m,any
equivalent form is also positive definite with arithmetic minimum greater or equal to
m, or equal to m, respectively. ⊓⊔
Proposition 29.5. R(m) is a generalized convex polyhedron.
Proof. Considering the definition of generalized convex polyhedra in Sect. 14.2, it
is sufficient to prove that, for any q ∈ bd R(m), there is a cube K with centre q in
E 1 2 d(d+1) such that K ∩ R(m) is a convex polytope. Let q ∈ bd R(m). The positive
quadratic form q has at most 2d − 1 pairs of minimum points, see Theorem 30.2.
If a cubic neighbourhood K of q is chosen sufficiently small, then K ⊆ P and the
minimum points of any form in K are amongst those of q. This means that
K ∩ R(m) = K ∩ � �
(a11,...,add) ∈ E 1 2 d(d+1) : �
�
aik uiuk ≥ m
u∈Z d
u minimum
point of q
is a polytope, concluding the proof. ⊓⊔
i,k