Gruber P. Convex and Discrete Geometry

198 Convex Bodies
Suppose now that any two points of M can be connected by a continuous curve of
finite length. Then the intrinsic or geodesic metric ϱ of the metric space 〈M,δ〉 is
defined as follows:
ϱ(x, y) is the infimum of the lengths of the continuous curves
in M connecting x, y for any x, y ∈ M.
In general the metrics δ and ϱ will be different, but they induce the same topology on
M. A continuous curve in M connecting two points x, y of M and of length ϱ(x, y)
is called a geodesic segment with endpoints x, y. It is non-trivial to show that, for
sufficiently differentiable closed convex surfaces, or, more generally, for sufficiently
smooth manifolds, this notion of geodesic segment coincides with the corresponding
differential geometric notion. For a hint to a proof, see Gruber [423].
Weyl’s Problem and Alexandrov’s Realization Theorem
The following version of Weyl’s [1020] problem was studied in convex geometry,
where a metric (d − 1)-sphere is a metric space homeomorphic to S d−1 .
Problem 10.3. Let M be a metric (d − 1)-sphere. Is there a closed convex surface
SinE d such that M and S are isometric, if both are endowed with their intrinsic
metric? In other words, can M be realized by a closed convex surface?
There exist many contributions to Weyl’s problem, mainly in differential geometry.
In the following we state without proof Alexandrov’s [15] solution in the context of
convex geometry.
Let M be a metric 2-sphere endowed with its intrinsic metric. If two geodesic
segments have a common endpoint it is possible to define a notion of lower angle
between them. The intrinsic metric on M is then said to have positive curvature if
each point of M has a neighbourhood with the following property: for each triangle
in this neighbourhood with geodesic segments as edges, the sum of the lower angles
between its edges is at least π. Alexandrov’s fundamental realization theorem now is
as follows.
Theorem 10.2. Let M be a metric 2-sphere with positive curvature. Then M is
isometric to a closed convex surface S in E 3 where both M and S are endowed
with their intrinsic metrics.
Rigidity of Convex Surfaces and Pogorelov’s Rigidity Theorem
The question arises, whether the convex surface S in Alexandrov’s realization theorem
is unique up to rigid motions? In other words, if S and T are closed convex
surfaces in E 3 which are isometric if both are endowed with their intrinsic metric, do
they coincide up to rigid motions? If this is the case, S is called rigid.
This problem for polytopal surfaces goes back to antiquity, see the discussion
in Sect. 17.1. In the context of differential geometry the first pertinent result seems