Gruber P. Convex and Discrete Geometry

Convex Bodies
Sporadic results on convex bodies have appeared in the mathematical literature since
antiquity, with an increasing rate in the nineteenth century. Systematic investigations
started only in the nineteenth and the early twentieth century with the work
of Cauchy, Steiner, Brunn and, in particular, Minkowski. Important contributors in
the twentieth century were Blaschke, Hadwiger, Alexandrov and many contemporary
mathematicians. The following quotation of Klee [593] shows roughly where,
in mathematics, this area is located and what are some of its characteristics:
The study of convex sets is a branch of geometry, analysis, and linear algebra that
has numerous connections with other areas of mathematics and serves to unify many
apparently diverse mathematical phenomena. It is also relevant to several areas of
science and technology.
During the twentieth century the relationship of convex geometry with analytic
flavor to other branches of mathematics and to applied areas increased greatly.
We mention differential and Riemannian geometry, functional analysis, calculus
of variations and control theory, optimization, geometric measure theory, inequalities,
Fourier series and spherical harmonics, probability, and mathematical physics.
Besides these relationships of a systematic character, there are minor connections
to numerous other areas, including complex function theory of one and several
variables, aspects of ordinary and partial differential equations, dynamical systems
and potential theory.
In this chapter, we try to justify the following observation of Ball [53]:
Although convexity is a simple property to formulate, convex bodies possess a surprisingly
rich structure.
We present the major analytic aspects of convex geometry together with many applications.
We begin with general properties of convex bodies, including some results
of combinatorial geometry. Then the boundary structure of convex bodies is investigated.
This comprises smooth, singular and extreme points. The natural topology
on the space of convex bodies is introduced next and Blaschke’s selection theorem
proved. Mixed volumes and quermassintegrals are treated in the following section.
The discussion of valuations is an important topic. Our exposition includes extension