Gruber P. Convex and Discrete Geometry

152 Convex Bodies
As a consequence of the Brunn–Minkowski theorem, we see that
(2) The function f (λ) = V � (1 − λ)C + λB d� 1 d − (1 − λ)V (C) 1 d − λV (B d ) 1 d
for 0 ≤ λ ≤ 1 with f (0) = f (1) = 0 is strictly concave, unless C is a ball,
in which case it is the zero function.
Since, by (1), f is differentiable for 0 ≤ λ ≤ 1, Proposition (2) shows that
(3) f ′ (0) ≥ 0, where equality holds if and only if C is a ball.
By (1) and (2),
f ′ (λ) = 1
d V � (1 − λ)C + λB d� 1 d −1� − d(1 − λ) d−1 V (C)
+ � − (d − 1)(1 − λ) d−2 λ + (1 − λ) d−1� S(C) + O(λ) �
Hence (3) implies that
or, equivalently,
+ V (C) 1 d − V (B d ) 1 d as λ → 0 + .
1
d V (C) 1 d −1� − dV(C) + S(C) � + V (C) 1 d − V (B d ) 1 d ≥ 0,
S(C)
dV(C) d−1
d
≥ V (B d ) 1 d = S(Bd )
dV(B d ) d−1 ,
d
where equality holds if and only if C is a ball. ⊓⊔
Proof (using Minkowski’s first inequality). Note that, for C ∈ Cp, wehaveS(C) =
dW1(C) = dV(B d , C ...,C). By Minkowski’s first inequality we have, V (B d ,
C,...,C) d ≥ V (B d )V (C) d−1 , where equality holds precisely in case where C is a
Euclidean ball. Taking into account that S(B d ) = dV(B d ) we thus obtain that
S(C) d
V (C) d−1 ≥ dd V (B d ) = dd V (B d ) d
V (B d ) d−1 = S(Bd ) d
V (B d ,
) d−1
where equality holds precisely in case where C is a ball. ⊓⊔
The Isodiametric Inequality
This inequality is due to Bieberbach [113]. We obtain it as a simple application of
the Brunn–Minkowski theorem. See also Sect. 9.2
Theorem 8.8. Let C ∈ C. Then
V (C) ≤
�
1
�d diam C V (B
2 d ),
where equality holds if and only if C is a solid Euclidean ball.