Gruber P. Convex and Discrete Geometry

10 Problems of Minkowski and Weyl and Some Dynamics 193
Then, there is a proper convex body C with area measure σC, unique up to translation,
such that
�
σC(B) =
dµd−1(u)
for each Borel set B ⊆ S
κ(u)
d−1 .
B
This equality shows that κ is the following Radon–Nikodym derivative.
κ = dµd−1
.
dσC
κ is called the generalized Gauss curvature of C. For the question whether or, more
precisely, when κ is the ordinary Gauss curvature of C, compare the references cited
in the remarks at the end of this section.
The extension of Alexandrov [12] and Fenchel and Jessen [335] of Minkowski’s
theorem is as follows.
Theorem 10.1. Let σ be a Borel measure on S d−1 . Then the following are equivalent:
(i) σ is not concentrated on a great circle of Sd−1 and
�
udσ(u) = o (componentwise).
S d−1
(ii) There is a proper convex body C, unique up to translation, with area measure σ .
Busemann [182], p.60, praised this result with the words:
... we have here a first example of a deeper theorem of differential geometry in the
large proved for a geometrically natural class of surfaces, i.e. without smoothness
requirements necessitated by the methods rather than the problem.
Proof. The proof rests on the corresponding result of Minkowski for convex polytopes,
see Theorem 18.2.
(i)⇒(ii) We first prove the existence of C. Byaspherically convex set on Sd−1 we mean the intersection of Sd−1 with a convex cone with apex o.Form = 1, 2,...,
decompose Sd−1 into finitely many pairwise disjoint spherically convex sets, each of
diameter at most 1/m. LetS1,...,Sn denote those among these sets which have
positive σ -measure. Here, and in the following, when i appears as an index it would
be better to write mi instead, but we do not do it in order to avoid clumsy notation.
Let
(15) ϱiui = 1
�
udσ(u),
σ(Si)
Si
where 0