Gruber P. Convex and Discrete Geometry

Proposition 26.2. E d /L is a compact Abelian topological group.
26 The Torus Group E d /L 397
Proof. Clearly, E d /L is Abelian. For the proof that E d /L is a topological group, we
have to show that the mapping
(1) (x, y) → x − y for x, y ∈ E d /L is continuous.
Let a, b ∈ E d /L and ε>0. Let x, y ∈ E d /L be contained in the ε/2-neighbourhoods
of a and b, respectively, i.e.
Choose l, m ∈ L such that:
�x − a� L , �y − b� L < ε
2 .
�x − a� L =�x − a − l�, �y − b� L =�y − b − m�.
The definition of �·� L then shows that
�x − y − (a − b)�L ≤�x − y − a + b − l + m�
≤�x − a − l�+�y − b − m� =�x− a�L +�y− b�L < ε ε
+ = ε.
2 2
Thus x − y is contained in the ε-neighbourhood of a − b, concluding the proof of (1).
Ed /L is a metric space. For the proof that it is compact, it is thus sufficient
to show that each sequence in Ed /L has a convergent subsequence. Let (xn) be a
sequence in Ed /L. LetFbe a fundamental parallelotope. Let xn ∈ F be such that
xn ∩ F ={xn}. This gives a sequence (xn) in the bounded set F ⊆ Ed . The Bolzano–
Weierstrass theorem then shows that the sequence (xn) contains a subsequence (xnk )
converging to a point x ∈ Ed , say. Let x = L + x. The definition of �·�L now implies
that
�xnk − x�L ≤�xnk − x� →0asn →∞.
Thus (xnk ) converges to x. ⊓⊔
Given a fundamental parallelotope F of L, define the distance modulo L on F by:
�x� L = inf � �x + l� :l ∈ L � =�x� L for x ∈ F.
�·� L yields a metric, and thus a topology, on F.LetF be endowed with this topology
(Fig. 26.2).
Proposition 26.3. The metric spaces 〈E d /L, �·� L 〉 and 〈F, �·� L 〉 are isometric. An
isometry is given by the mapping
x → x ∩ Fforx∈ E d /L.
Proof. Left to the reader. ⊓⊔