Gruber P. Convex and Discrete Geometry

Steiner’s formula in d − 1 dimensions then shows that
6 Mixed Volumes and Quermassintegrals 109
v � (C + λB d )|u ⊥� = v � C|u ⊥ + λB d |u ⊥�
�d−1
� �
d − 1
=
wi(C|u
i
⊥ )λ i .
i=0
Since C|u ⊥ varies continuously with u and wi(·) is continuous, we may integrate
over Sd−1 and Cauchy’s surface area formula, applied to C + λB d , shows that
(4) S(C + λB d �d−1
� � �
d − 1 1
) =
wi(C|u
i
⊥ ) dσ(u)λ i for λ ≥ 0.
i=0
κd−1
Sd−1 Since S(C + λB d ) = dW1(C + λB d ), Steiner’s formula for W1(·) yields the following,
see Theorem 6.14:
(5) S(C + λB d �d−1
� �
d − 1
) = d
Wi+1(C)λ
i
i for λ ≥ 0.
i=0
Finally, equating coefficients in (4) and (5), implies (3). ⊓⊔
A Remark on the Proofs
In the proofs of Steiner’s and Kubota’s formulae for quermassintegrals, we have
expressed the same quantity in two different ways as polynomials. Hence these polynomials
must be identical, i.e. corresponding coefficients coincide. This finally yields
the desired formulas. Expressing the same quantity in different ways and equating is
a common method of proof in integral geometry, see Santaló [881].
Why are Quermassintegrals or Mean Projection Measures Called So?
Iterating (3) with respect to the dimension, yields the general formulae of Kubota.
These express the quermassintegrals Wi(C) of C as the mean of the (d − i)dimensional
volumes of the projections of C onto (d − i)-dimensional linear subspaces.
These volumes are called Quermaße in German.
Blaschke’s Problem for Quermassintegrals
We conclude this section with the following major problem which goes back to
Blaschke [125].
Problem 6.1. Determine the set �� W0(C), . . . , Wd−1(C) � : C ∈ C � ⊆ E d .
In case d = 2 the solution is given by the isoperimetric inequality: The set in question
is {(P, A) : P 2 ≥ 4π A, P, A ≥ 0}, where P and A stand for perimeter and area.
The problem is open for d ≥ 3. For some references, see Hadwiger [468], Schneider
[907], and Sangwine-Yager [877, 878].