Gruber P. Convex and Discrete Geometry

174 Convex Bodies
Proof. Let ε>0. By the sphericity theorem, there are hyperplanes H1,...,Hn1 ,
through o, such that
�
V (C)
st Hn ···st
1
H1C ⊆ (1 + ε)
Again, by the sphericity theorem, there are hyperplanes Hn1+1,...,Hn2 , through o,
such that
while maintaining
κd
� 1 d
B d .
st Hn ···st
2
Hn1 +1 (st �
V (D)
Hn ···st
1
H1 D) ⊆ (1 + ε)
st Hn ···st
2
Hn1 +1 (st �
V (C)
Hn ···st
1
H1C) ⊆ (1 + ε)
Applying the same argument with ε/2 to the convex bodies st Hn 2 ···st H1 C,
st Hn 2 ···st H1 D ∈ Cp, etc., finally yields the corollary. ⊓⊔
Remark. Considering the theorem and its corollary, the question arises, how fast
do suitable iterated Steiner symmetrals of C approximate ϱB d ? The first result in
this direction was proved by Hadwiger [463]. A more precise estimate is due to
Bourgain, Lindenstrauss and Milman [160]. There are an absolute constant α>0
and a function α(ε) > 0 such that the following statement holds: Let C be a convex
body C of volume V (B d ) and ε>0. Then, by at most α d log d + α(ε) d Steiner
symmetrizations of the body or the polar body, one obtains a convex body D with
(1 − ε)B d ⊆ D ⊆ (1 + ε)B d .
For a pertinent result involving random Steiner symmetrizations, see Mani-Levitska
[683]. A sharp isomorphic result is due to Klartag and Milman [589]. A lower bound
for the distance of iterated Steiner symmetrals of C from ϱB d was given by Bianchi
and Gronchi [112]. See [660, 878] for additional references.
Blaschke Symmetrization
Blaschke [124], p.103, introduced the following concept of symmetrization. Given
a convex body C and a hyperplane H in E d ,letC H be the (mirror) reflection of C
in H. Then the Blaschke symmetrization of C with respect to H is the convex body
1 1
C +
2 2 C H .
In recent years the Blaschke symmetrization has also been called Minkowski
symmetrization. A surprisingly sharp sphericity result for this symmetrization is due
to Klartag [588].
κd
κd
� 1 d
� 1 d
B d ,
B d .