Gruber P. Convex and Discrete Geometry

10 Problems of Minkowski and Weyl and Some Dynamics 199
to be Liebmann’s [657] theorem on the rigidity of spheres within the class of
sufficiently smooth closed convex surfaces. In his proof the following result, already
conjectured by Minding, is shown first. A sufficiently smooth closed convex surface
with constant Gauss curvature is a sphere. Then he makes use of the fact that
the Gauss curvature is determined by the intrinsic metric. For some references, see
the book by Su [976]. We point out the remarkable and totally unexpected result
of Nash [766] and Kuiper [621] that a sufficiently smooth closed convex surface is
C 1 -diffeomorphically isometric to a topological sphere which is folded such as to
form a set of arbitrarily small (Euclidean) diameter. It is this Nash who won the
Nobel prize in economics in 1995, see Milnor [727].
The final rigidity result for closed convex surfaces in convex geometry is the
following rigidity theorem of Pogorelov [803]:
Theorem 10.3. Let S and T be two closed convex surfaces in E 3 which are isometric
with respect to their intrinsic metrics. Then S and T are congruent.
So far no simple proof of this result is available. Surveys of the relevant literature
are due to Pogorelov [805], Su [976] and Ivanova-Karatopraklieva and Sabitov [538,
539].
10.3 Evolution of Convex Surfaces and Convex Billiards
While, in a majority of results in convex geometry, fixed objects, in particular fixed
convex bodies are studied, there are several groups of results in the last decades
which deal with moving objects, for example with deformation of surfaces. Of these
investigations with a dynamical aspect we mention the following:
flexible polytopal spheres and frameworks
evolution of convex surfaces by curvature driven flows
billiards
In the following we first cite some results on evolution of closed convex surfaces
by flows which are driven by the mean and the Gauss curvature. Then caustics of
convex billiard tables are considered. No proofs are given.
For more information on rigidity and flexibility of closed convex surfaces,
polytopal spheres and frameworks, see Sects. 10.2, 17.1 and 17.2 and the references
given there.
Evolution of Convex Curves and Surfaces by Curvature Driven Flows
All convex curves and surfaces considered in the following are assumed to be
sufficiently differentiable.
If S0 is a closed convex curve in E 2 , the problem is to find a family {St : t ≥ 0}
of closed convex curves given by a sufficiently differentiable function x(s, t) where,
for fixed t, the expression x(s, t) is an arc-length parametrization of St such that, for
fixed s, the point x(s, t) moves in time t in the direction of the inner unit normal