Gruber P. Convex and Discrete Geometry

128 Convex Bodies
and the Bi have pairwise disjoint interiors. We may then represent conv A in the form
conv A = B1 ∪···∪ Bm ∪ P1 ∪···∪ Pn where Pj ∈ P
and such that the boxes and convex polytopes B1,...,Pn have pairwise disjoint
interiors. Thus (4) and (1) imply the following:
φ(A) = φ(B1) +···+φ(Bm) ≤ φ(B1) +···+φ(Bm) + φ(P1) +···+φ(Pn)
= φ(conv A) ≤ φ(C),
concluding the proof of (5).
The next step is to prove the following counterpart of (5):
(6) φ(C) ≤ φ(B) for C ∈ C, B ∈ L(B), where C ⊆ int B.
Given C, B, choose P ∈ P, such that C ⊆ P ⊆ B. This is possible since C ⊆ int B.
Represent B in the form
B = B1 ∪···∪ Bm, where Bi ∈ B
and the Bi have pairwise disjoint interiors. Then (1), (4) and (1) again yield the
following:
φ(C) ≤ φ(P) = φ � P ∩ (B1 ∪···∪ Bm) � = φ � (P ∩ B1) ∪···∪(P ∩ Bm) �
= φ(P ∩ B1) +···+φ(P ∩ Bm) ≤ φ(B1) +···+φ(Bm) = φ(B),
concluding the proof of (6).
In the last part of the proof we show that
(7) φ(C) = cV(C) for C ∈ C.
Let C ∈ C. Then (3), (5), (6) and (3) show that
sup{cV(A) : A ∈ L(B), A ⊆ C} ≤φ(C)
≤ inf{cV(B) : B ∈ L(B), C ⊆ int B}.
Since C is Jordan measurable, by Theorem 7.4, we have
sup{V (A) : A ∈ L(B), A ⊆ C} =V (C)
= inf{V (B) : B ∈ L(B), C ⊆ int B}.
Hence φ(C) = cV(C), concluding the proof of (7) and thus of the theorem. ⊓⊔
Second Characterization of the Volume of Convex Bodies
Much more difficult than the proof of the first characterization of the volume is the
proof of Hadwiger’s characterization of the volume, see [462, 464, 468], where a
rigid motion is proper if it has determinant 1.