Gruber P. Convex and Discrete Geometry

212 Convex Bodies
the maximum of the diameters of the facets of Pn tends to 0 as n → ∞.This,
together with (5), implies the following:
(10) δV (C, Pn) ≥ � � �
volume of the subset of Pn below Ji
for all sufficiently large n.
i
Let Fnik, k = 1,...,mni, be the facets of Pn below C such that F ′ nik
Clearly,
(11) mni →∞as n →∞,
(12) mn1 +···+mnl ≤ n for all sufficiently large n.
∩ J ′
i �= ∅.
Let snik ∈ relint C ′ be the projection into Hi of the point where Fnik touches C.
Then the
(13) volume of the subset of Pn below Ji
= �
k
�
F ′ nik∩J ′
i
� fi(s) − fi(snik) − grad fi(snik) · (s − snik) � ds.
Since fi is of class C 2 , the remainder term in Taylor’s formula shows that the
integrand here is
1
2 qsnik+ξ(s−snik)(s − snik) for s ∈ F ′ ′
nik ∩ J i
with suitable ξ ∈[0, 1] depending on s. Since the maximum of the diameters of the
facets of Pn tends to 0 as n →∞, (5) shows that, for sufficiently large n, thesets
F ′ nik which meet J ′
i are all contained in U ′
i . For such n, it follows from (6) that the
integrand is bounded below by
1
2λ qi(s − snik) for s ∈ F ′ ′
nik ∩ J i .
Thus (10), (13), the lower bound for the integrand in (13), (11), (3), (2), (8), (12) and
(9) yield (4), where summation on i is from 1 to l and on k from 1 to mi:
δ V (C, P c (n) ) = δV (C, Pn) ≥ 1 � �
�
qi(s − snik) ds
2λ
≥ 1
2λ
≥ 1
2λ
≥ δ
2λ 2
�
�
i
�
i
�
i
J ′
i
i
k
F ′ nik∩J ′
i
min {qi(s − snik)} ds
k=1,...,mni
inf
S⊆H
#S=m ni
�
J ′
i
min
t∈S {qi(s − t)} ds
v(J ′
d+1
i ) d−1 (det qi) 1
d−1
1
m 2
d−1
ni