Gruber P. Convex and Discrete Geometry

192 Convex Bodies
halfspace {z : un · z ≤ un · xn}. Now, noting that Cn → C, un → u ∈ F ⊆ Sd−1 ,
and xn → x ∈ bd C, the halfspace {z : u · z ≤ u · x} is a support halfspace of C at x.
Thus x ∈ n −1
C (u) ⊆ n−1
C (F), concluding the proof of (10).
It follows from (4), (9) and (10) that
� � −1
lim sup µd−1 ϱ nCn n→∞
(F)�� � −1
≤ µd−1 nC (F) � .
Since
� −1
µd−1 nC (S d−1 ) � � � −1
= µd−1 ϱ nCn (Sd−1 ) �� = µd−1(bd C),
� −1
the measures µd−1 ϱ(nCn (· )� are probability measures on Sd−1 up to some constant.
� � −1
Thus (6) implies that the measures µd−1 ϱ nCn (· )�� converge weakly to the measure
� −1
µd−1 nC (·) � . Taking into account (7), this implies that the area measures σCn (· ) =
� −1
µd−1 nCn (· )� � −1
converge weakly to the area measure σC(· ) = µd−1 nC (· ) � . ⊓⊔
Corollary 10.1. Let C, D ∈ Cp. Then
V (C, D,...,D) = 1
�
d
S d−1
hC(u) dσD(u).
Proof. Choose convex polytopes Pn ∈ Pp, n = 1, 2,..., such that Pn → D. By
Proposition 10.2 the area measures σPn
Hence, in particular:
converge weakly to the area measure σD.
�
�
(11) hC(u) dσPn (u) → hC(u) dσD(u).
S d−1
S d−1
Lemma 6.5 and the definition of the area measure of polytopes show that
(12) V (C, Pn,...,Pn) = 1
d
�
hC(u F)v(F) = 1
d
�
hC(u) dσPn (u).
F facet of Pn
According to Theorem 6.8 mixed volumes are continuous in their entries. Since
Pn → D, we thus have:
(13) V (C, Pn,...,Pn) → V (C, D,...,D).
The corollary is now an immediate consequence of Propositions (11)–(13). ⊓⊔
Alexandrov’s and Fenchel–Jessen’s Generalization of Minkowski’s Theorem
Minkowski [736, 739] proved the following result: Let κ : Sd−1 → R + be a continuous
function
�
such that
u
(14)
κ(u) dµd−1(u) = o (componentwise).
S d−1
S d−1