Gruber P. Convex and Discrete Geometry

21 Lattices 359
triangle conv{o,(u,v),(x, y)} contains no point of Z 2 , except its vertices. Considering
its mirror image in o and the translation of the latter by the vector (u,v)+ (x, y),
we see that the parallelogram generated by (u,v)and (x, y) contains no point of Z 2 ,
except its vertices. The parallelogram F ={α(u,v)+ β(x, y) : 0 ≤ α, β < 1} then
contains only the point o of Z 2 . Given a point l ∈ Z 2 , we may subtract integer multiples
of (u,v)and (x, y) from it such that the resulting lattice point is contained in
F and thus must coincide with o. Hence l is an integer linear combination of (u,v)
and (x, y). Thus (u,v),(x, y) form a basis of the lattice Z 2 . By Theorem 21.1 their
determinant is ±1 and by our choice of (x, y) it is positive. Hence uy − vx = 1. ⊓⊔
21.2 Characterization of Lattices
Since the notion of lattice is important for many purposes, it is sometimes useful to
have at hand an alternative description.
This section contains a simple characterization of lattices. This result or, more
precisely, its proof will be used in Proof of Theorem 21.3. It is also a tool for the
Venkov–McMullen theorem 32.3 on characterization of parallelohedra.
A Characterization of Lattices
A subset of E d is called discrete if any bounded set contains only finitely many of
its points or, equivalently, if it has no point of accumulation. A characterization of
lattices based on the notions of group and discrete set now is as follows.
Theorem 21.2. Let L ⊆ E d . Then the following statements are equivalent:
(i) L is a lattice.
(ii) L is a discrete sub-group of E d which is not contained in a hyperplane.
Proof. (i)⇒(ii) Let {b1,...,bd} be a basis of L. Ifl, m are integer linear combinations
of b1,...,bd, then so is l − m. Hence L is a sub-group of E d . For the proof
that L is discrete, note that
(1) � α1b1 +···+αdbd :−1 0 be the radius of a ball with centre at o which is contained in the open
parallelotope in (1). Then the distance from o to any point of L\{o} is at least ϱ.
Therefore, we have �l − m� ≥ϱ for l, m ∈ L, l �= m.IfL is not discrete, it contains
a bounded infinite subset. This subset then has at least one accumulation point. Any
two distinct points of this subset, which are sufficiently close to the accumulation
point, have distance less than ϱ. This contradiction concludes the proof that L is
discrete. L is not contained in a hyperplane since it contains the points o, b1,...,bd.
(ii)⇒(i) It is sufficient to show the following:
(2) There are d linearly independent vectors b1,...,bd ∈ L such that L =
� u1b1 +···+udbd : ui ∈ Z � .