Gruber P. Convex and Discrete Geometry

232 Convex Bodies
What Does the Boundary of a Typical Convex Body Look Like?
An easy proof will yield the following answer.
Theorem 13.1. Most proper convex bodies are smooth and strictly convex.
Proof. Smoothness: For n = 1, 2,...,let
Cn = � C ∈ Cp :∃p ∈ bd C, u,v ∈ S d−1 such that
�u − v� ≥ 1
n , C ⊆{x : x · u ≤ p · u}, {x : x · v ≤ p · v}� .
Simple compactness arguments show that
(1) Cn is closed in Cp.
(It is sufficient to show that C1, C2, ···∈Cn, C ∈ Cp, C1, C2, ···→C implies that
C ∈ Cn too.) To see that
(2) int Cn =∅,
assume the contrary. Since the smooth convex bodies are dense in Cp,thesetCn then
would contain a smooth convex body, but this is incompatible with the definition of
Cn. (1) and (2) imply that Cn is nowhere dense. Hence
∞�
Cn is meagre.
n=1
To conclude the proof of the smoothness assertion, note that
Strict convexity: Replacing Cn by
� C ∈ Cp : C is not smooth � =
∞�
Cn.
n=1
Dn = � C ∈ Cp :∃p, q ∈ bd C :�p − q� ≥ 1
n , [p, q] ⊆bd C� ,
the proof is similar to the proof in the smoothness case. ⊓⊔
Remark. This result has been refined and generalized in the following directions.
Most convex bodies are not of class C 1+ε , see Gruber [423] (ε = 1) and Klima
and Netuka [599] (ε >0).
Most convex bodies are of class C 1 , but have quite unexpected curvature properties.
For a multitude of pertinent results, mainly due to Zamfirescu, see the
surveys [431, 1041].
Zamfirescu [1040] proved that all convex bodies are of class C 1 and strictly convex,
with a countable union of porous sets of exceptions. A porous set is meagre
but the converse does not hold generally. For a definition see [431].