Gruber P. Convex and Discrete Geometry

40 Convex Bodies
results, characterizations of volume and Hadwiger’s functional theorem. The latter
is applied to prove the principal kinematic formula of integral geometry. A central
theme is the Brunn–Minkowski inequality, which leads to geometric and physical
isoperimetric inequalities and to the concentration of measure phenomenon. The
following section deals with Steiner symmetrization and Schwarz rearrangement
which are valuable tools, for example for isoperimetric inequalities of mathematical
physics. Area measures and the intrinsic metric of convex surfaces are then
studied, including the existence and uniqueness problems of Minkowski and Weyl.
We present solutions of these problems due to Alexandrov, Fenchel and Jessen, and
Pogorelov. Then we give some hints to dynamical aspects of convex geometry dealing
with the evolution of convex surfaces and billiards. Next come John’s ellipsoid
theorem and the reverse isoperimetric inequality. Then asymptotic best approximation
of convex bodies is treated and applied to the isoperimetric problem for polytopes.
Special convex bodies have always attracted interest. Here, simplices, balls
and ellipsoids are considered. Finally, the space of convex bodies is studied from
topological, measure, metric, group and lattice viewpoints.
Applications deal with complex function theory of several variables, Lyapunov’s
convexity theorem for vector-valued measures, Pontryagin’s minimum principle,
Birkhoff’s convexity theorem on doubly stochastic matrices, a series of results from
mathematical physics, in particular the theorem of Wulff on the form of crystals, and
Choquet’s characterization of vector lattices.
In this chapter we will often use convex polytopes, related notions and their simple
properties, in particular approximation properties. The reader who is not familiar
with convex polytopes may consult the introductory sections of the next chapter.
The reader who wants to get more detailed information is referred to the books
and surveys of Blaschke [124], Bonnesen and Fenchel [149], Alexandrov [18],
Eggleston [290], Hadwiger [466, 468], Santaló [881], Leichtweiss [640], Burago
and Zalgaller [178], Schneider [907], Groemer [405], Thompson [994], Gardner
[359, 360], Klain and Rota [587], Ball [53] and Magaril-Il’yaev and Tikhomirov
[678]. In addition, we refer to parts I and IV of the Handbook of Convex Geometry
[475], to Convexity and Its Applications [219] and to the collected or selected
works of Minkowski [745], Blaschke [129] and Alexandrov [18, 19].
There are omissions. The theory of curvature and area measures will only be
mentioned briefly. More, but by no means sufficient material deals with the local
theory of normed spaces. For thorough representations of these areas, see Schneider
[907], respectively, Pisier [802], Tomczak-Jaegermann [1001], Ball [53], and the pertinent
surveys and chapters in the Handbook of Convex Geometry [475], the Handbook
of the Geometry of Banach Spaces [477] and the monograph of Benyamini and
Lindenstrauss [97]. Integral geometry and geometric probability are only touched.
For information see Santaló [881] and Schneider and Weil [911].
3 Convex Sets, Convex Bodies and Convex Hulls
In this section the notions of convex sets, convex bodies and convex hulls are introduced.
Several simple properties are presented, including Carathéodory’s theorem on
convex hulls. Next, a short excursion into combinatorial geometry will include the