Gruber P. Convex and Discrete Geometry

y (9), (4) and (7), or
� �
ci(mC) − ci(C) � Wi(D) = 0forD∈ C.
i
7 Valuations 139
An argument that was used twice before, then shows that ci(mC) = ci(C), concluding
the proof of (12).
Having proved (10)–(12), Hadwiger’s functional theorem shows that
(13) ci(·) = ci0W0(·) +···+cidWd(·) for C ∈ C
with suitable coefficients cij.
In the last step of the proof, the coefficients cij in (13) will be determined. This
is done by applying (3), (9) and (13) to the special convex bodies λB d ,µBd , noting
that ν is normalized and using Theorem 6.13 (ii):
φ(λB d ,µB d �
) = V � λB d − µ rB d� �
dν(r) = V � (λ + µ)B d� dν(r)
O(d)
O(d)
= V � (λ + µ)B d� = (λ + µ) d V (B d �
� �
d
) = κd λ
i
i
i µ d−i
= �
cijWi(λB d ) W j(µB d ) = �
cijWi(B d ) W j(B d )λ d−i d− j
µ
i, j
Hence:
�d� κd i
(14) cij = 0 for i + j �= d and ci d−i =
Wi(B d ) Wd−i(B d ) =
since Wi(B d ) = κd, by Steiner’s theorem on the volume of parallel bodies 6.6.
Propositions (3), (9), (13) and (14) settle the theorem for C.
To prove the theorem also for L(C), we proceed as follows: given a convex body
C ∈ C and a rigid motion m, the valuations
�
χ(C ∩ m · ) dµ(m) and Wd−i(·)
M
on C are continuous and invariant with respect to proper rigid motions. By Groemer’s
extension theorem 7.3, these valuations have unique extensions to L(C). Since their
values for D1 ∪···∪ Dm ∈ L(C), where D j ∈ C, are determined by the inclusion–
exclusion formula, (2) continues to hold for the given C and all D ∈ L(C). Thus
(2) holds for all C ∈ C and D ∈ L(C). Then, given D ∈ L(C), a similar argument
shows that (2) holds for all C ∈ L(C) and the given D ∈ L(C). Thus (2) holds for all
C ∈ L(C) and D ∈ L(D), concluding the proof of the theorem. ⊓⊔
Remark. For extensions of the principal kinematic formula to homogeneous spaces,
see Howard [523] and Fu [345]. A comprehensive survey of more recent results
on kinematic and Crofton type formulae in integral geometry is due to Hug and
Schneider [528].
i, j
�d� i
κd