Gruber P. Convex and Discrete Geometry

The second property of φ says that
(5) φ(·, D) is a valuation on C for given D ∈ C.
Let B, C ∈ C such that B ∪ C ∈ C. Since
(B ∪ C) ∩ mD = (B ∩ mD) ∪ (C ∩ mD),
(B ∩ C) ∩ mD = (B ∩ mD) ∩ (C ∩ mD),
and since χ is a valuation on C, we obtain the equality
7 Valuations 137
χ � (B ∪ C) ∩ mD � + χ � (B ∩ D) ∩ mD � = χ(B ∩ mD) + χ(C ∩ mD)
for m ∈ M. Integrating this equality over M then shows that
φ(B ∪ C, D) + φ(B ∩ C, D) = φ(B, D) + φ(C, D),
concluding the proof of (5). Properties (4) and (5) yield the third property of φ, which
says that
(6) φ(C, ·) is a valuation on C for given C ∈ C.
The left invariance of the Haar measure µ yields the following fourth property of φ.
(7) φ(C, mD) = φ(C, D) for C, D ∈ C, m ∈ M.
More intricate is the proof of the fifth and last property,
(8) φ(·, ·) is continuous on C × C.
To see this, note that
φ(C, D) =
�
O(d)
V (C − rD) dν(r)
by (3). V (C − rD) is continuous in (C, D, r), using the natural topology on O(d)
(matrix norms). Since O(d) is compact, integration yields a function which is continuous
in (C, D), concluding the proof of (8).
For given C ∈ C the function φ(C, ·) is a continuous valuation on C by (6) and
(8). By (7) it is invariant with respect to proper rigid motions. An application of
Hadwiger’s functional theorem thus shows that
(9) φ(C, D) = c0(C)W0(D) +···+cd(C)Wd(D) for C, D ∈ C
with suitable coefficients c0(C), . . . , cd(C).
We now investigate these coefficients and show first that
(10) ci(·) is a valuation on C for i = 0,...,d.