Gruber P. Convex and Discrete Geometry

9 Symmetrization 175
9.2 The Isodiametric, Isoperimetric, Brunn–Minkowski, Blaschke–Santaló
and Mahler Inequalities
If C is a proper convex body but not a ball, then a suitable Steiner symmetral of C
has smaller isoperimetric quotient than C. This observation led Steiner [958, 959],
erroneously, to believe that Steiner symmetrization shows that balls have minimum
isoperimetric quotient. The gap in this proof of the isoperimetric inequality was filled
by Blaschke. He showed that Steiner symmetrization together with the Blaschke
selection theorem, or rather the sphericity theorem of Gross, also provide easy proofs
of the isodiametric, the isoperimetric and the Brunn–Minkowski inequalities.
Below we first present the proofs of Gross [407] and Blaschke [124] of these
inequalities, but do not discuss the equality cases. Then the proof of Ball [47] and
Meyer and Pajor [720] of the Blaschke–Santaló inequality will be given, again,
excluding a discussion of the equality case. As a counterpart of the Blaschke–Santaló
inequality, we present an inequality of Mahler. Let κd = V (B d ).
The Isodiametric Inequality
A simple proof using d Steiner symmetrizations yields the following result of
Bieberbach [113], a refinement of which was given by Urysohn [1004]. For a
different proof, see Sect. 8.3.
Theorem 9.2. Let C ∈ C. Then V (C) ≤ 1
2 d (diam C)d V (B d ).
Proof. By symmetrizing C with respect to each coordinate hyperplane we obtain a
convex body D which is symmetric in the origin o and with V (D) = V (C) and
diam D ≤ diam C, see Proposition 9.1(vi,viii). Then D ⊆ ( 1 2 diam D)Bd . ⊓⊔
The Isoperimetric Inequality
An equally simple proof yields the next result.
Theorem 9.3. Let C ∈ Cp. Then
S(C) d
V (C) d−1 ≥ S(Bd ) d
V (B d .
) d−1
Proof. By Corollary 9.1, there are hyperplanes H1, H2,... through o, such that
st Hn ···st H1C →
� V (C)
Now using Proposition 9.1(vii) and the continuity of the surface area S(·), see
Theorem 6.13, we obtain the desired inequality:
�� � 1
V (C) d
S(C) ≥ S(st H1C) ≥···→ S
B d
�
= V (C) d−1
d
S(B d ). ⊓⊔
κd
κd
� 1 d
B d .
κ d−1
d
d