Gruber P. Convex and Discrete Geometry

108 Convex Bodies
Third, let C be a convex body of dimension d − 1. We may assume that C ⊆ v⊥ for suitable v ∈ Sd−1 . Then
�
1
v(C|u ⊥ ) dσ(u) = 1
�
|u · v| dσ(u)v(C) = 2v(C) = S(C).
κd−1
Sd−1 κd−1
Sd−1 If, fourth, dim C < d − 1, then both sides in (1) are 0 and thus coincide. ⊓⊔
An Integral-Geometric Interpretation of Cauchy’s Formula
Given a set L of lines in E d , a natural measure for L can be defined as follows: For
any (d − 1)-dimensional subspace u ⊥ of E d , where u ∈ S d−1 , consider the lines
in L which are orthogonal to u ⊥ .Letv(u) denote the measure of the intersection
of this set of lines with u ⊥ (if the intersection is measurable). The integral of v(u)
over S d−1 with respect to the ordinary surface area measure, then, is the measure
of the set L of lines (if the integral exists). Clearly, this measure is rigid motion
invariant. Cauchy’s surface area formula now says that the surface area of a convex
body C, that is the area of its boundary bd C, equals (up to a multiplicative constant)
the integral of the function which assigns to each line the number of its intersection
points with bd C. This interpretation extends to all sufficiently smooth surfaces in
E d and is the starting point for so-called integral geometric surface area. Formore
on integral geometry, see the standard monograph of Santaló [881] and Sect. 7.4.
Geometric measure theory is treated by Falconer [317] and Mattila [696].
Kubota’s Formulae for Quermassintegrals
Let wi(·), i = 0,...,d −1, denote the quermassintegrals in d −1 dimensions. Then
one may write Cauchy’s formula in the following form:
W1(C) = 1
�
w0(C|u
d κd−1
⊥ ) dσ(u).
S d−1
It was the idea of Kubota [619] to extend this formula to all quermassintegrals.
We state the following special case.
Theorem 6.16. Let C ∈ C. Then
(3) Wi(C) = 1
�
wi−1(C|u
d κd−1
⊥ ) dσ(u) for i = 1,...,d.
Proof. First note that
S d−1
(C + λB d )|u ⊥ = C|u ⊥ + λB d |u ⊥ for u ∈ S d−1 .