Gruber P. Convex and Discrete Geometry

142 Convex Bodies
8.1 The Classical Brunn–Minkowski Inequality
The Brunn–Minkowski inequality was first proved by Brunn [173,174] with a clever
but rather vague proof. The equality case, in particular, was not settled satisfactorily,
as was pointed out by Minkowski. Then, both Brunn [175] and Minkowski [735]
provided correct proofs. At present several essentially different proofs are known, as
well as a series of far-reaching extensions and applications.
Dinghas [270] commented on Brunn’s proof as follows:
The ingenious idea of Brunn to relate two convex sets by equal volume ratios ... saw
substantial progress in the last twenty years. It did not yield only refinements of old,
but produced also new results.
The proof that will be given below is a variant of the proof of Brunn [173, 174],
respectively of its precise version due to Kneser and Süss [600]. It is by induction
and makes use of the idea of Brunn which relates the d-dimensional case to
the (d − 1)-dimensional case by equal volume ratios: let u ∈ Sd−1 and H(t) =
{x : u · x = t}. For0≤s ≤ 1lettC(s) and tD(s) be such that the hyperplanes
H(tC(s)) and H(tD(s)) divide the volume of C, respectively, D, in the ratio s : 1−s.
Clearly,
C + D ⊇ � � �
C ∩ H tC(s) � + D ∩ H � tD(s) �� .
0≤s≤1
Now apply induction to C ∩ H � tC(s) � + D ∩ H � tD(s) � and use Fubini’s theorem.
A different proof is due to Blaschke [124]. It makes use of a property of Steiner
symmetrization, namely that
st(C + D) ⊇ st C + st D.
This makes it possible to reduce the proof to the trivial case where C and D are balls.
For Blaschke’s proof, see Sect. 9.2.
The Classical Inequality
Our aim is to prove the Brunn–Minkowski inequality:
Theorem 8.1. Let C, D ∈ C. Then:
(1) V (C + D) 1 d ≥ V (C) 1 d + V (D) 1 d ,
where equality holds if and only if one of C, D is a point, or C and D are improper
and lie in parallel hyperplanes, or C and D are proper and positive homothetic.
Proof. The following inequality will be used below:
(2) � v 1
d−1 + w 1 � �
d−1 V W
�
d−1 + ≥ (V
v w
1 d + W 1 d ) d for v,w,V, W > 0,
where equality holds if and only if
v
V d−1
d
= w
W d−1 .
d