Gruber P. Convex and Discrete Geometry

5 The Boundary of a Convex Body 71
each complete metric or locally compact space is Baire. The rational numbers with
the usual topology are an example of a space which is not Baire.
Ever since their introduction by Baire [45] and Osgood [780], Baire categories
have been applied successfully in real analysis. See, e.g. Oxtoby [782] and Holmes
[520]. In recent years they found numerous applications in convex geometry, compare
the surveys of Zamfirescu [1041] and Gruber [431], and Sect. 13.1.
Given a proper convex body, its boundary is compact and thus Baire by the category
theorem. The finite-dimensional case of the density theorem of Mazur [701] is
as follows.
Theorem 5.2. Let C ∈ Cp. Then the set of singular points of bd C is meagre in bd C.
Proof. For n = 1, 2,...,let
Sn = � x ∈ bd C : there are support hyperplanes at x with angle at least 1 �
.
n
A simple compactness argument shows that Sn is closed in bd C. In order to show
that
(5) Sn is nowhere dense in bd C,
assume the contrary. Being closed, Sn then contains a relatively open subset G of
bd C.IfB is a solid Euclidean ball of sufficiently small radius, there is a translate of
B contained in C which touches bd C at a point x ∈ G. Then, clearly, C has a unique
support hyperplane at x. Since this is in contradiction to x ∈ G ⊆ Sn, the proof of
(5) is complete. Clearly,
∞�
(6) Sn = � x ∈ bd C : x singular � .
n=1
The theorem now follows from (5) and (6). ⊓⊔
Differentiability
A (d − 1)-dimensional manifold in E d may be represented explicitly, implicitly or
by means of parameters. The problem arises to move differentiability properties or
results, from one representation to the others. In differential geometry this is done by
means of refined versions of the theorems on implicit and inverse functions, using
differentiability assumptions. Unfortunately, in general, such possibilities are not
available in convex geometry. Thus one has to argue more carefully, even in simple
cases. In the following we consider the problem of transferring first and secondorder
differentiability at given points from convex functions to convex bodies and
vice versa.
First-Order Differentiability
Let D be an open convex set in E d and f : D → R a convex function. Clearly,
notions such as support hyperplanes and regular and singular boundary points and