Gruber P. Convex and Discrete Geometry

6 Mixed Volumes and Quermassintegrals 105
including the equality case, see Theorem 8.7. Note that, if C has dimension d − 1,
then S(C) equals twice the (d −1)-dimensional volume of C.Ifd = 2, we also write
P(C) for S(C) and call P(C) the perimeter of C.
We point out that this notion of surface area coincides with the elementary surface
area of proper convex polytopes and with the common notion of area of differentiable
surfaces in E d . For more complicated sets it is – if it exists – in general
larger than common notions of surface area such as the Lebesgue or the Hausdorff
surface area or the more recent notions of integral geometric surface areas, perimeter
and currents. Such questions are not discussed in this book and we refer to Burago
and Zalgaller [178] and Morgan [756].
Properties of Quermassintegrals
In Theorem 6.13 we collect results for quermassintegrals, proved in Sect. 6.3 for the
more general case of mixed volumes.
Theorem 6.13. The following statements hold for i = 0,...,d:
(i) Wi(·) is rigid motion invariant and thus, in particular, translation invariant,
i.e. Wi(mC) = Wi(C) for C ∈ C and each rigid motion m of E d .
(ii) Wi(·) is (positive) homogeneous of degree d − i, i.e. Wi(λC) = λ d−i Wi(C)} for
C ∈ C and λ ≥ 0.
(iii) Wi(·) is continuous on C.
(iv) Wi(·) is non-decreasing on C with respect to set inclusion.
(v) Wi(·) is a valuation on C,thatis,ifC, D ∈ C are such that C ∪ D ∈ C,
then
Wi(C ∪ D) + Wi(C ∩ D) = Wi(C) + Wi(D).
Proof. Properties (i)–(iv) are immediate consequences of Propositions 6.5, 6.6 and
Theorems 6.9, 6.10. Property (v) follows from the first proof of Theorem 6.10 on
putting D2 = ··· = Dd = B d , λ2 = ··· = λd = 1 and comparing the coefficients
of λ i . ⊓⊔
Steiner Formulae for Quermassintegrals
Steiner’s formula 6.6 for the volume of parallel bodies can be extended as follows.
Theorem 6.14. Let C ∈ C . Then
Wi(C + λB d �d−i
� �
d − i
) =
Wi+k(C)λ
k
k for λ ≥ 0 and i = 0,...,d.
In particular,
k=0
S(C + λB d ) = dW1(C + λB d �d−1
� �
d − 1
) = d Wk+1(C)λ
k
k for λ ≥ 0.
k=0