Gruber P. Convex and Discrete Geometry

214 Convex Bodies
(20) mni ≥ 1
λ τin for all sufficiently large n
(21) mn1 +···+mnl ≤ n
Choose points snik ∈ L ′ i , k = 1,...,mni, such that
�
(22) min {qi(s − snik)} ds = inf
k=1,...,mni
S⊆L ′ �
i
#S=mni min
t∈S {qi(s − t)} ds.
L ′ i
Let tnik be the point on the lower part of bd C which projects onto snik. For sufficiently
large n, the points tnik are distributed rather densely over bd C. Thus the
intersection of the support halfspaces of C at these points is a convex polytope, say
Qn, where Qn is circumscribed to C and – by (21) – has at most n facets. Clearly,
Qn → C as n →∞. Then (15) implies that
(23) δ V (C, Qn) ≤ �
i
≤ �
�
The expression in {·}equals
i
� �
volume of the subset of Qn below Li
L ′ i
L ′ i
�
min fi(s) − fi(snik)
k=1,...,mni
for all sufficiently large n.
− grad fi(snik) · (s − snik) � ds
1
2 qsnik+ξ(s−snik)(s − snik)
with suitable ξ ∈[0, 1], which by (17) and (6) is at most
λ
2 qi(s − snik).
Hence (23), (19), (3), (16), (20), (17) and (8), (18) and (15) show (14):
δ V (C, P c (n) ) ≤ δV (C, Qn) ≤ λ �
�
min {qi(s − snik)}ds
2 k=1,...,mni
≤ λ2 δ
2
≤
�
i
d+1 2
λ2+ d−1 + d−1 δ
2
i
L ′ i
v(L ′ d+1
i ) d−1 (det qi) 1
d−1
�
i
1
m 2
d−1
ni
v(K ′ d+1
i ) d−1 (det qi) 1
d−1
1
τ 2
d−1
i
1
n 2
d−1