Gruber P. Convex and Discrete Geometry

8 The Brunn–Minkowski Inequality 153
Proof. We may suppose that V (C) > 0. We consider two cases. First, let C be
centrally symmetric. Without loss of generality, we may assume that o is the centre
of C. Then C ⊆ ( 1 2 diam C)Bd and thus
V (C) ≤
�
1
�d diam C V (B
2 d ),
where equality holds if and only if C = ( 1 2 diam C)Bd .
Second, let C be not centrally symmetric. The Brunn–Minkowski theorem 8.2
then implies that
and thus:
V (C) 1 d = 1
2 V (C) 1 d + 1
2 V (− C) 1 �
1
� 1
d 1
d < V (C − C) =
2 2 V (C − C) 1 d ,
(4) V (C) < 1
V (C − C).
2d Since C − C is symmetric:
�
1
(5) V (C − C) ≤ diam(C − C)
2
by the first case. Next note that
(6)
� d
V (B d )
diam(C − C) = max{�(u − v) − (x − y)� :u,v,x, y ∈ C}
≤ max{�u − x� :u, x ∈ C}+max{�v − y� :v, y ∈ C}
= 2diamC.
Now combine (4)–(6) to see that
V (C) <
�
1
�d diam C V (B
2 d ). ⊓⊔
The isodiametric inequality can be used to show that d-dimensional Hausdorff measure
coincides, up to a multiplicative constant, with the Lebesgue measure in E d ,see
Morgan [756].
Urysohn’s inequality
A refinement of the isodiametric inequality is the following inequality of Urysohn
[1004] where, instead of the diameter, the mean width is used:
Theorem 8.9. Let C ∈ Cp. Then
V (C) ≤
�
1
2 w(C)
�d V (B d ),
where equality holds if and only if C is a solid Euclidean ball.