Gruber P. Convex and Discrete Geometry

188 Convex Bodies
Both problems influenced the development of differential geometry and convex
geometry throughout the twentieth century. An important additional aspect is that
of the uniqueness of the convex body C and of the convex surface S, respectively. In
the context of convex geometry, both problems have been solved satisfactorily at the
right level of generality, including the uniqueness problems. Many of the pertinent
results go back to Alexandrov and Pogorelov in the context of Alexandrov’s differential
geometry of not-necessarily differentiable convex surfaces.
In the following, the notion of area measure of a convex body will be introduced
and, using this, the solution of Alexandrov [12] and Fenchel and Jessen [335] of
Minkowski’s problem, including uniqueness, will be presented. Then we consider
the notion of intrinsic metric and state, without proofs, Alexandrov’s [15] realization
and Pogorelov’s [803] uniqueness or rigidity theorem which together solve Weyl’s
problem for general convex surfaces in E 3 . The known proofs of these results are
long and full of tedious technical details.
For more information on the above problems, the reader is referred to the books
of Busemann [182], Pogorelov [806] and Schneider [907], to the surveys of Ivanova-
Karatopraklieva and Sabitov [538, 539] and the references below.
In addition to investigations of fixed convex bodies or classes of convex bodies,
there are results in convex geometry of a dynamical type, dealing with rigidity and
deformation of closed convex surfaces, convex surfaces with boundary, non-convex
polytopal spheres and frameworks. For rigidity we refer to Sects. 10.2, 17.1 and
17.2. Early results on the evolution of convex polytopal spheres are related to crystal
growth, see the references in Sect. 8.4 dealing with Wulff’s theorem. In recent years
numerous results on deformation of closed convex surfaces by flows of different
types have been studied in the context of differential geometry and partial differential
equations. In some cases the surfaces shrink to a point and before collapsing
their form becomes more and more spherical. These contributions may be considered
convexity results, at least in some cases.
A different dynamical aspect in convex geometry is provided by billiards.
Billiards have been studied in mathematics at least since Birkhoff and have attracted
a lot of interest in the context of dynamical systems. There are results dealing with
periodic and dense trajectories, with caustics and with the behaviour induced by
boundary points of the billiard tables with particular curvature properties. Several of
these investigations also belong to convex geometry.
In the following we try to convey some of the dynamical flavor of evolution
and billiards by considering evolution of closed convex surfaces by curvature driven
flows and convex caustics in billiard tables.
10.1 Area Measure and Minkowski’s Problem
In this section, we define and investigate the notion of (surface) area measure of convex
bodies. Then, given a Borel measure σ on S d−1 with certain properties, existence
and uniqueness of a convex body with area measure σ is proved. This is then used
to define Blaschke addition of proper convex bodies. We follow Alexandrov [10],
Pogorelov [806] and Schneider [907].