Gruber P. Convex and Discrete Geometry

Clearly,
6 Mixed Volumes and Quermassintegrals 79
(4) A d × d permutation matrix is not a convex combination of other d × d
doubly stochastic matrices.
The theorem now is a consequence of Propositions (1), (2) and (4). ⊓⊔
6 Mixed Volumes and Quermassintegrals
Schneider [907] describes the Brunn–Minkowski theory in the preface of his monograph
with the following words:
Aiming at a brief characterization of the Brunn–Minkowski theory, one might say
that it is the result of merging two elementary notions for point sets in Euclidean
space: vector addition and volume.
The original problem of the Brunn–Minkowski theory is to obtain information on the
volume
V (λC + µD)
of the Minkowski linear combination λC + µD ={λx + µy : x ∈ C, y ∈ D}
of two convex bodies C, D for λ, µ ≥ 0, in terms of information on C and D.
Prior to Minkowski, the only results of what is now the Brunn–Minkowski theory
were Cauchy’s [198] surface area formula, Steiner’s [959] formula for the volume
of parallel bodies of a convex body, the proof of the isoperimetric inequality by
Schwarz [922] and the inequality of Brunn [173, 174] (–Minkowski). These results
then were considered as interesting, but rather isolated contributions to geometry.
Their fundamental importance became visible only after Minkowski [744] built,
around them, a voluminous theory, now called after Brunn and himself. A central
notion is that of mixed volumes. Important later contributors to this theory were
Hadwiger [468] and Alexandrov [18]. Of contemporary mathematicians we mention
Schneider [907]. The Brunn–Minkowski theory deals, amongst others, with
mixed volumes, the corresponding measures, with developments around the Brunn–
Minkowski inequality, geometric inequalities and other topics of an analytic flavour
in convex geometry. Its numerous applications and relations to other areas range
from isoperimetric problems of various sorts, including isoperimetric inequalities of
mathematical physics, to crystallography, statistics and algebraic geometry.
In this section, we first give basic notions and preliminary results dealing with
Minkowski addition and the Hausdorff metric. Then Blaschke’s selection theorem is
presented. Next, we consider Minkowski’s theorem on mixed volumes and Steiner’s
formula for the volume of parallel bodies and study properties of mixed volumes
and quermassintegrals, including Minkowski’s inequalities, Cauchy’s surface area
formula and Kubota’s formulae for quermassintegrals.
The notions and properties of volume and convex polytopes will be used several
times in this section but will be treated in detail only in Sect. 7 and in the chapter
“Convex Polytopes”.