Gruber P. Convex and Discrete Geometry

6 Mixed Volumes and Quermassintegrals 103
Hadwiger will be treated, in which the quermassintegrals are characterized as special
valuations.
For more information we refer to Leichtweiss [640], Schneider [905, 908],
Sangwine-Yager [878] and McMullen and Schneider [716].
Quermassintegrals and Intrinsic Volumes
Recall Steiner’s theorem on parallel bodies, Theorem 6.6: Given a convex body C ∈
C,wehave,
V (C + λB d � �
� �
d
d
) = W0(C) + W1(C)λ +···+ Wd(C)λ
1
d
d for λ ≥ 0,
where the coefficients
Wi(C) = V (C,...,C,
B
� �� �
d−i
d ,...,B d
), i = 0,...,d,
� �� �
i
are the quermassintegrals of C.IfC is contained in a subspace of E d , one can define
the quermassintegrals of C both in E d and in this subspace. Unfortunately the result
is not the same as the following proposition shows, where v(·) is the volume and
wi(·), i = 0,...,d − 1, are the quermassintegrals in d − 1 dimensions, κ0 = 1, and
κi, i = 1,...,d, is the i-dimensional volume of B i .
Proposition 6.7. Let C ∈ C(E d−1 ) and embed E d−1 into E d as usual (first d − 1
coordinates). Then
Proof. Let u = (0,...,0, 1). Then
d�
i=0
� d
i
=
i κi
Wi(C) = wi−1(C) for i = 1,...,d.
d κi−1
�
Wi(C)λ i = V (C + λB d ) =
�λ
−λ
i=0
�λ
−λ
v � C + (λ 2 − t 2 ) 1 2 B d−1� dt =
−λ
v � (C + λB d ) ∩ (E d−1 + tu) � dt
�λ
−λ
�
� �
d − 1
wi(C)(λ
i
2 − t 2 ) i �
2 dt
� d−1
�d−1
� � �λ
d − 1
=
wi(C) (λ
i
2 − t 2 ) i �d−1
� �
d − 1
2 dt =
wi(C)
i
κi+1
λ
κi
i+1
for λ ≥ 0
by Steiner’s formula in E d , Steiner’s formula in E d−1 and Fubini’s theorem which
is used to calculate the (i + 1)-dimensional volume of the ball λB i+1 . Now equate
coefficients. ⊓⊔
i=0
i=0