Gruber P. Convex and Discrete Geometry

x smooth
5 The Boundary of a Convex Body 69
x + N(x)
x singular
Fig. 5.1. Smooth and singular points, normal cones
Using the dimension of the normal cone, one may classify the singular boundary
points (Fig. 5.1).
The natural question arises, to determine the size of the set of singular boundary
points of C.
The Hausdorff Measure of the Set of Singular Points
A continuous curve of finite length λ in Ed can be covered by a sequence of Euclidean
balls of arbitrarily small diameters such that the sum of the diameters is arbitrarily
close to λ. This property was used by Carathéodory [190] to define the linear
measure of more general sets. His idea was extended by Hausdorff [482] as follows:
For 0 ≤ s ≤ d, thes-dimensional Hausdorff measure µs(A) of a set A ⊆ Ed is
defined by
�
� �∞
µs(A) = lim inf (diam Un)
ε→+0
s : Un ⊆ E d ∞� �
, diam Un ≤ ε, A ⊆ Un
�
,
n=1
where for U ⊆ E d ,diamU = sup{�x − y� :x, y ∈ U} is the diameter of U. The
above defined µs, actually, is not a measure but an outer measure. If A is Lebesgue
measurable then, up to a constant depending on d, its Hausdorff measure µd(A)
equals the Lebesgue measure µ(A) or V (A) of A.IfA is a Borel set in the boundary
of a given proper convex body C, then µd−1(A) equals, up to a constant depending
on d, the ordinary Lebesgue or Borel area measure σ(A) or S(A) of A. IfK is
a Jordan curve in E d , then µ1(K ) is its length. For more detailed information on
measure theory, respectively, geometric measure theory see, e.g. Falconer [317] or
Mattila [696].
Anderson and Klee [28] gave the following result, where a set has σ -finite measure
if it is a countable union of sets of finite measure.
Theorem 5.1. Let C ∈ Cp. Then the set of singular points of bd C has σ -finite
(d − 2)-dimensional Hausdorff measure.
Proof. In a first step we show the following.
(1) Let x ∈ bd C. Then NC(x) is a closed convex cone with apex o.
To see that NC(x) is a convex cone with apex o, letu,v ∈ NC(x) and λ, µ ≥ 0.
Then u · y ≤ u · x and v · y ≤ v · x for all y ∈ C. Thus (λu + µv) · y ≤ (λu + µv) · x
n=1