Gruber P. Convex and Discrete Geometry

11 Approximation of Convex Bodies and Its Applications 211
Let U be the set on the lower part of bd C which projects onto U ′ . Clearly, U is a
neighbourhood of p in bd C.
The following inequality is well known.
(2)
� �
1
m
σ d+1
d−1
1
d+1
d−1
+···+σm �� d−1
d+1
≥ 1
m (σ1 +···+σm) for σ1,...,σm > 0.
By means of a suitable linear transformation, Zador’s Theorem 33.2 for α = 2
yields the following asymptotic formula:
(3) Let J ⊆ H be Jordan measurable with v(J) >0 and q a positive definite
quadratic form on H. Then
�
min{q(s
− t)} ds ∼ δv(J)d+1 d−1 (det q)
t∈S 1 1
d−1 as m →∞,
inf
S⊆H
#S=m
J
where δ>0 is a constant depending only on d.
After these preparations, the first step is to show that
(4) δ V (C, P c (n) ) ≥
δ
⎛
⎝
�
κC(x) 1
⎞
d+1 dσ(x) ⎠
2λ 4d
d−1
for all sufficiently large n.
bd C
d+1
d−1
m 2
d−1
1
n 2
d−1
The open neighbourhoods U considered above cover the compact set bd C. Thus
there is a finite subcover. By Lebesgue’s covering lemma (see, e.g. [572], p.154),
each set of sufficiently small diameter in bd C is then contained in one of the neighbourhoods
of the subcover. Thus we may choose finitely many small pieces Ji,
i = 1,...,l, in bd C, points pi and neighbourhoods Ui of pi in bd C, support hyperplanes
Hi of C at pi, convex functions fi, and quadratic forms qu = qpi u, qi = qpi pi
as in the preparations, such that the following statements hold:
(5) The sets Ji are compact and pairwise disjoint, and their projections
J ′
i ⊆ Ui ⊆ relint C ′ are Jordan measurable
(6) 1
λ qi(s) ≤ qu(s) ≤ λ qi(s) for s ∈ Hi, u ∈ U ′
i
(7) 1
λ det qi ≤ det qu ≤ λ det qi for u ∈ U ′
i
(8) 1
λ κC(u) ≤ det qi ≤ λκC(u) for u ∈ U ′
i
(9) �
�
i
J ′
i
κC(u) 1
d+1 du ≥ 1
λ
�
bd C
κC(x) 1
d+1 dσ(x)
Let Pn ∈ P c (n) , n = d + 1,..., be a sequence of best approximating convex
polytopes of C. Since δ V (C, Pn) → 0 and C is strictly convex (note that κC > 0),