Gruber P. Convex and Discrete Geometry

6 Mixed Volumes and Quermassintegrals 89
a polynomial in the coefficients of the linear combination. The coefficients of the
polynomial are the mixed volumes. The following quotation from Hilbert’s [502],
p.XIX, obituary on Minkowski illustrates the meaning of mixed volumes.
...Thus the concept of mixed volume appears as the simplest generalization which
comprises the notions of volume, surface area, total mean curvature as special cases.
In this way the latter notions are related much more closely to each other. Thus we
may expect now to obtain a deeper understanding than was possible before, of the
mutual relations of these notions...
An important special case is Steiner’s [959] formula for the volume of parallel
bodies. The coefficients of the corresponding polynomial are, up to multiplicative
constants, the so-called quermassintegrals. Mixed volumes and quermassintegrals
played a dominant role in convex geometry throughout the twentieth century.
In this section we prove Minkowski’s theorem on mixed volumes and Steiner’s
formula. In the following sections the notions of mixed volumes and quermassintegrals
will be investigated in more detail.
For references, see the introduction of this chapter.
Minkowski’s Theorem on Mixed Volumes
Theorem 6.5. Let C1,...,Cm ∈ C. Then there are coefficients V (Ci1 ,...,Cid ),
1 ≤ i1,...,id ≤ m, called mixed volumes, which are symmetric in the indices and
such that
V (λ1C1 +···+λmCm) =
m�
i1,...,id =1
V (Ci1 ,...,Cid )λi1 ···λid for λ1,...,λm ≥ 0.
The proof is by induction. It follows a clear line and, basically, is not difficult. To
make it more transparent, it is split into several steps. The first tool is Lemma 6.1
above.
A convex polytope is the convex hull of a finite set in E d .LetP denote the space
of all convex polytopes in E d .GivenP ∈ P,aface of P is the intersection of P with
a support hyperplane. It is the convex hull of the intersection of the finite set which
determines P with the support hyperplane and thus also a convex polytope. A face
of dimension d − 1 is called a facet. For more information, see Sect. 14.1
Lemma 6.2. Let P1,...,Pm ∈ P. Then the following claims hold:
(i) P = λ1 P1 +···+λm Pm ∈ P for λ1,...,λm ≥ 0.
(ii) There is a finite set U ⊆ S d−1 such that for all λ1,...,λm ≥ 0, forwhich
P = λ1 P1 +···+λm Pm is a proper convex polytope, the exterior unit normal
vectors of the facets of P are contained in U.
Proof. (i) We may assume that λ1,...,λm > 0, otherwise consider fewer polytopes.
Then the following will be shown.
(1) Let e ∈ P be extreme and e = λ1e1 +···+λmem, where ei ∈ Pi. Then
each ei is extreme in Pi.