Gruber P. Convex and Discrete Geometry

172 Convex Bodies
Proposition 9.2. Let C ∈ Cp be o-symmetric. Then, Steiner symmetrization, with
respect to a given hyperplane through o, satisfies the following inequality:
V � (st C) ∗� ≥ V (C ∗ ).
Proof. We may suppose that this hyperplane is E d−1 ={x : xd = 0}. Then
st C = �� x, 1
2 (s − t)� : (x, r), (x, s) ∈ C � ,
C ∗ = � (y, t) : x · y + st ≤ 1for(x, s) ∈ C � ,
(st C) ∗ = � (w, t) : x · w + 1
2 (r − s)t ≤ 1for(x, r), (x, s) ∈ C� .
For a set A ⊆ E d and t ∈ R, letA(t) ={v ∈ E d−1 : (v, t) ∈ A}. Then
1�
�
∗ ∗
C (t) + C (−t)
2
= �1
(y + z) : x · y + rt ≤ 1, w· z + s(−t) ≤ 1for(x, r), (w, s) ∈ C�
2
⊆ �1
(y + z) : x · y + rt ≤ 1, x · z + s(−t) ≤ 1for(x, r), (x, s) ∈ C�
2
⊆ �1
2
(y + z) : x · 1
2
1
(y + z) + (r − s)t ≤ 1for(x, r), (x, s) ∈ C�
2
= � w : x · w + 1
(r − s)t ≤ 1for(x, r), (x, s) ∈ C�
2
= (st C) ∗ (t) for t ∈ R.
Since C is o-symmetric, C∗ is also o-symmetric. Thus C∗ (t) =−C∗ (−t) and therefore
v � C∗ (t) � = v � C∗ (−t) � . The Brunn–Minkowski inequality in d − 1 dimensions
then shows that
v � (st C) ∗ (t) � �
1�
� ∗ ∗
≥ v C (t) + C (−t) �
≥
2
�
1
2 v�C ∗ � 1 1
(t) d−1 +
2 v� C ∗ (−t)
� 1 �d−1 d−1
= v � C ∗ (t) � for t ∈ R.
Now, integrating over t, Fubini’s theorem yields V � (st C) ∗� ≥ V (C ∗ ). ⊓⊔
The Sphericity Theorem of Gross
The following highly intuitive sphericity theorem of Gross [407] and its corollary
were used by Blaschke [124] for easy proofs of the isodiametric, the isoperimetric
and the Brunn–Minkowski inequalities. Let κd = V (B d ).
Theorem 9.1. Let C ∈ Cp with V (C) = V (B d ). Then, there is a sequence
C1, C2,...of convex bodies, each obtained from C by finitely many successive
Steiner symmetrizations with respect to hyperplanes through o, such that
C1, C2, ···→ B d .