Gruber P. Convex and Discrete Geometry

The Isoperimetric Inequality for Convex Bodies
8 The Brunn–Minkowski Inequality 151
In the following we present three related proofs of the isoperimetric inequality for
convex bodies. All three proofs make use of Minkowski’s notion of surface area, see
Sect. 6.4. The first proof is a simple application of the Brunn–Minkowski inequality.
It yields the isoperimetric inequality, but does not settle the equality case. The
second proof, which is due to Minkowski [738], is based on the Brunn–Minkowski
inequality, including the equality case, and Steiner’s formula for the volume of parallel
bodies. It shows when there is equality. It is curious to note that Brunn [173],
p.31, thought that
This theorem [the Brunn–Minkowski theorem] cannot be used for a proof of the
maximal property of the ball.
The third proof makes use of Minkowski’s first inequality and also settles the equality
case. Since the proof of Minkowski’s first inequality is based on the Brunn–
Minkowski theorem, it is not surprising, that the isoperimetric inequality for convex
bodies can be derived from it.
The isoperimetric inequality for convex bodies is as follows.
Theorem 8.7. Let C ∈ Cp(E d ). Then
S(C) d
V (C) d−1 ≥ S(Bd ) d
V (B d ,
) d−1
where equality holds if and only if C is a solid Euclidean ball.
Proof (using the Brunn–Minkowski theorem). The Brunn–Minkowski theorem
shows that
V (C + λB d ) 1 d ≥ V (C) 1 d + λV (B d ) 1 d for λ>0.
Thus
For λ → 0, we get
V (C + λB d ) − V (C)
λ
S(C) ≥ V (C) d−1
d
≥ dV(C) d−1
d V (B d ) 1 d + O(λ) as λ → 0.
dV(Bd )
V (B d ) d−1
S(C)
, or
d
d
V (C) d−1 ≥ S(Bd ) d
V (B d . ⊓⊔
) d−1
Proof (using the Brunn–Minkowski theorem and Steiner’s theorem for the volume
of parallel bodies). By Steiner’s theorem:
(1) V � (1 − λ)C + λB d� is a positive polynomial in λ for 0 ≤ λ ≤ 1, and
V � (1 − λ)C + λB d� = (1 − λ) d V (C) + (1 − λ) d−1 λS(C) + O(λ 2 )
as λ →+0.