Gruber P. Convex and Discrete Geometry

hold for all sufficiently large n. Hence
Cn ⊆ C + ε
�
C = 1 +
ϱ ε
�
C,
�
ϱ
1 − ε
�
C +
ϱ
ε
ϱ C = C ⊆ Cn + ε
ϱ C.
7 Valuations 123
Using support functions and arguing as in the proof of the cancellation law for convex
bodies (Theorem 6.1), we obtain the inclusions
�
1 − ε
� �
C ⊆ Cn ⊆ 1 +
ϱ
ε
�
C
ϱ
for all sufficiently large n. This, together with (8), yields the following:
�
1 − ε
�d �
V (C) ≤ V (Cn) ≤ 1 +
ϱ
ε
�d V (C).
ϱ
For given ε>0 and all sufficiently large n,
which, in turn, implies (10) for V (C) >0. Assume second, that V (C) = 0. Since
C1, C2, ···→C, for any ε>0
Cn ⊆ C + εB d
for all sufficiently large n. Thus, by Steiner’s theorem on the volume of parallel
bodies 6.6,
V (Cn) ≤ V (C + εB d ) = V (C) + O(ε) = O(ε)
for given ε>0 and all sufficiently large n.
This proves (10) for V (C) = 0. The proof of (10), and thus of (9), is complete.
It remains to show that
(11) V is rigid motion invariant.
Taking into account the definition of V , it is sufficient to show that any two congruent
boxes have the same volume. To see this, note that, by Lemma 7.1, it is sufficient to
prove the following, where, by a dissection of a proper convex polytope, we mean a
representation of it as a union of finitely many proper convex polytopes with pairwise
disjoint interiors:
(12) Let P, T be two congruent rectangles in E 2 . Then P can be dissected into
finitely many convex polygons such that suitable translations of these polygons
form a dissection of T .
In other words, P and T are equidissectable with respect to (the group of) translations
(Fig. 7.1). Since equidissectability is transitive, as can be shown easily, the following
argument leads to (12): first, P is equidissectable to a parallelogram Q, one pair of
edges of which is parallel to a pair of edges of T . Q is equidissectable to a rectangle,
the edges of which are parallel to those of T . R is equidissectable to a rectangle S,
one side of which has length at least the length l of the corresponding parallel edges
of T and at most twice this length. S and T are equidissectable. We omit the details.
⊓⊔