Gruber P. Convex and Discrete Geometry

10 Problems of Minkowski and Weyl and Some Dynamics 191
Since F0 is compact, it is the intersection of the decreasing sequence (Nm), where
Hence
Nm =
�
F0 + 1
m Bd� ∩ bd C.
(5) µd−1(F0) = lim sup µd−1(Nm).
m→∞
Since each Nm is a neighbourhood of F0 in bd C, we see that for given m we have,
Fn ⊆ Nm for all sufficiently large n. Therefore
lim sup µd−1(Fn) ≤ µd−1(Nm) for m = 1, 2,...
n→∞
This together with (5) yields (4).
The second tool is the following special case of the Portmanteau theorem, see
Bauer [82].
(6) Let τ,τ1,τ2,..., be Borel probability measures on S d−1 . Then the following
statements are equivalent:
(i) τ1,τ2,... converge weakly to τ.
(ii) lim sup τn(F) ≤ τ(F) for each compact set F ⊆ S
n→∞
d−1 .
For the proof of Proposition 10.2 we may suppose that o ∈ int C and o ∈
int Cn for n = 1, 2,... Let ϱ : E d \{o} →bd C be the radial mapping of E d \{o}
onto bd C with centre o. Since o ∈ int C and C1, C2, ··· → C, it is not difficult to
show that
(7) The mappings ϱ : bd Cn → bd C and their inverses are Lipschitz with
Lipschitz constants converging to 1.
We will apply (6). Let F ⊆ Sd−1 be compact. By (1) and since ϱ is continuous,
the sets ϱ � n −1
Cn (F)� ⊆ bd C are compact. By Blaschke’s selection theorem for compact
sets, we may assume by considering a suitable subsequence and renumbering,
if necessary, that
(8) lim
n→∞
and
σCn (F) exists and equals the limit superior of the original sequence,
(9) ϱ � n −1
Cn (F)� → F0, say, where F0 is compact in bd C.
In order to show that
(10) F0 ⊆ n −1
C (F) ⊆ bd C,
let x ∈ F0. By (9) there are points yn ∈ ϱ � n −1
Cn (F)� ⊆ bd C with yn → x. Choose
xn ∈ n −1
Cn (F) ⊆ bd Cn such that ϱ(xn) = yn. By (7) and since yn → x, and
ϱ(x) = x, we have that xn → x. Letun∈ F such that xn ∈ n −1
Cn (un). By considering
a suitable subsequence and renumbering, if necessary, we may suppose that
un → u ∈ Sd−1 , say. Since F is compact, u ∈ F. Cn is contained in the support