Gruber P. Convex and Discrete Geometry

136 Convex Bodies
�
χ(C ∩ mD) dµ(m)
M
�
χ(B ∩ mD) dµ(m)
M
may be interpreted as the probability that a congruent copy of D which meets B
also meets C. There are similar interpretations for χ replaced by the quermassintegrals
Wi.
Our aim is to prove the principal kinematic formula:
Theorem 7.10. Let C, D ∈ L(C). Then
(2)
�
d�
χ(C ∩ mD) dµ(m) =
1
M
κd
i=0
� �
d
Wi(C) Wd−i(D).
i
The quermassintegrals Wi, including χ, are continuous valuations on C, see Theorem
6.13. Hence they can be extended in a unique way to valuations on L(C) by
Groemer’s extension theorem. These extensions are applied in (2).
Proof. We consider, first, the case C. Define a function φ : C × C → R by
(3) φ(C, D) = µ �� m ∈ M : C ∩ mD �= ∅ �� �
= χ(C ∩ mD) dµ(m)
=
=
�
O(d)
�
O(d)
M
� �
χ � C ∩ (rD+ t) � �
dt dν(r)
E d
V (C − rD) dν(r) for C, D ∈ C,
noting that {t : C ∩ (rD+ t) �= ∅}=C − rD. Since {m ∈ M : C ∩ mD �= ∅}is
compact in M = O(d)×E d and thus measurable, φ is well defined. In order to apply
the functional theorem of Hadwiger to φ, several properties of φ have to be shown
first.
The first property of φ is the following:
(4) φ(C, D) = φ(D, C) for C, D ∈ C.
χ is invariant with respect to rigid motions and the Haar measure µ is unimodular,
that is, if in an integrand the variable m is replaced by m −1 , the integral does not
change its value. See, e.g. [760], p.81. Thus (4) can be obtained as follows:
�
�
φ(C, D) = χ(C ∩ mD) dµ(m) = χ(m −1 C ∩ D) dµ(m)
M
�
M
= χ(D ∩ m −1 �
C) dµ(m) = χ(D ∩ mC) dµ(m) = φ(D, C).
M
M