Gruber P. Convex and Discrete Geometry

Steiner’s Formula for Parallel Bodies
6 Mixed Volumes and Quermassintegrals 93
C
C + λB 2
Fig. 6.2. Steiner’s formula for parallel bodies
An important special case of Theorem 6.5 is the following result of Steiner [959]
(see Fig. 6.2).
Theorem 6.6. Let C ∈ C. Then
V (C + λB d � �
d
) = W0(C) + W1(C)λ +···+
1
where
� �
d
Wd(C)λ
d
d forλ ≥ 0,
Wi(C) = V (C,...,C,
B
� �� �
d−i
d ,...,B d
), i = 0,...,d.
� �� �
i
The polynomial on the right side is the Steiner polynomial and the quantities Wi are
the quermassintegrals or the mean projection measures of the convex body C.
Remark. It is a natural, yet not intensively studied problem to extract geometric
properties of convex bodies from their Steiner polynomials. An open conjecture of
Teissier [993] relates the roots of the Steiner polynomial to the inradius and the
circumradius of the corresponding convex body. For a series of interesting geometric
results related to Steiner polynomials in the 2- and 3-dimensional case, see
Hernández Cifre and Saorín [496] who also give some references to the literature.
Remark. For an explanation why quermassintegrals are called quermassintegrals,
see a remark after Kubota’s theorem 6.16.
6.3 Properties of Mixed Volumes
Mixed volumes are a seminal notion in convex geometry and, thus, have been
investigated intensively from various viewpoints. Mixed area measures are localized
versions of mixed volumes. Mixed volumes form a bridge between convex and
algebraic geometry. For hints in this direction, see Sect. 19.5.