Gruber P. Convex and Discrete Geometry

134 Convex Bodies
Mean Width and the (d-1)st Quermassintegral
As a simple consequence of Hadwiger’s functional theorem we derive the following
identity:
Corollary 7.1. Wd−1(C) = κd
w(C) for C ∈ C, where w(C) is the mean width
2
of C,
w(C) = 2
�
hC(u) dσ(u).
d κd
S d−1
Proof. The main step of the proof is to show that
(20) w(·) is a valuation.
To see this, let C, D ∈ C be such that C∪D ∈ C. In the second proof of Theorem 6.10
we showed the following equalities for the support functions of C ∩ D and C ∪ D:
(21) hC∩D = min{hC, h D} and hC∪D = max{hC, h D}.
Proposition (20) can now be obtained as follows:
w(C) + w(D)
= 2
�
(hC(u) + h D(u)) dσ(u)
d κd
= 2
d κd
S d−1
�
S d−1
min{hC(u), h D(u)} dσ(u) + 2
d κd
= w(C ∩ D) + w(C ∪ D)
�
S d−1
max{hC(u), h D(u)} dσ(u)
by (21), concluding the proof of (20).
w is a valuation by (20). It is easy to see that it is continuous, rigid motion
invariant and positive homogeneous of degree 1. Hadwiger’s functional theorem then
shows that w is a multiple of Wd−1. NowletC = B d and note that dWd−1(B d ) =
d κd by Steiner’s formula and w(B d ) = 2 to determine the factor. ⊓⊔
7.4 The Principal Kinematic Formula of Integral Geometry
The following quote from Santaló [879], preface, gives an idea of what integral
geometry is all about:
To apply the idea of probability to random elements that are geometric objects (such
as points, lines, geodesics, congruent sets, motions, or affinities), it is necessary, first,
to define a measure for such sets of elements. Then, the evaluation of this measure
for specific sets sometimes leads to remarkable consequences of a purely geometric
character, in which the idea of probability turns out to be accidental. The definition
of such a measure depends on the geometry with which we are dealing. According
to Klein’s famous Erlangen Program (1872), the criterion that distinguishes one