Gruber P. Convex and Discrete Geometry

70 Convex Bodies
for all y ∈ C and therefore λu + µv ∈ NC(x). To see that NC(x) is closed, let
u1, u2, ··· ∈ NC(x), where un → u ∈ E d , say. Then un · y ≤ un · x for all y ∈ C.
Since un → u, we obtain u · y ≤ u · x for all y ∈ C. Hence u ∈ NC(x), concluding
the proof of (1).
Note that the following statement holds:
(2) Let x ∈ bd C. Then x is singular if and only if dim NC(x) ≥ 2.
Next, let S be the countable set of all simplices in E d \C of dimension d − 2 with
rational vertices and let pC : E d → C denote the metric projection of E d onto C,
see Sect. 4.1. For the proof that
(3) � x ∈ bd C : x singular � ⊆ � � pC(S) : S ∈ S � ,
let x ∈ bd C be singular. Then dim NC(x) ≥ 2 by (2). Thus we may choose a simplex
S ∈ S with NC(x) ∩ S �= ∅. Then x ∈ pC(S), concluding the proof of (3).
Since by Lemma 4.1 the metric projection pC is non-expansive, the simple property
of the Hausdorff measure that non-expansive mappings do not increase the
Hausdorff measure, implies that
�
(4) µd−2 pC(S) � ≤ µd−2(S)