Gruber P. Convex and Discrete Geometry

Diameter and Circumradius: Jung’s Theorem.
3 Convex Sets, Convex Bodies and Convex Hulls 49
The following estimate of Jung [555] relates circumradius and diameter. We reproduce
a proof which makes use of Helly’s theorem.
The circumradius of a set A in E d is the minimum radius of a Euclidean ball
which contains the set. It is easy to see that the ball which contains A and whose
radius is the circumradius is unique. It is called the circumball of A. Thediameter
diam A of A is the supremum of the distances between two points of the set.
Theorem 3.3. Let A ⊆ Ed be bounded. Then A is contained in a (solid Euclidean)
ball of radius
�
d
� 1
2
ϱ =
diam A.
2d + 2
Proof. In the first step we show that
(1) The theorem holds for sets A consisting of d + 1 or fewer points.
Let A be such a set and c the centre of a ball containing A and of minimum radius,
say σ . We may assume that c = o. Let
{x1,...,xn} ={x ∈ A :�x� =σ }, where n ≤ d + 1.
Then c ∈ conv{x1,...,xn}, since otherwise we could decrease σ by moving c (= o)
closer to conv{x1,...,xn}. Hence
Then
c = o = λ1x1 +···+λnxn, where λ1,...,λn ≥ 0, λ1 +···+λn = 1.
1 − λk =
=
n�
i=1
i�=k
λi ≥
n�
i=1
i�=k
λi
1
(diam A) 2
� � �n
2
2σ − 2
Now, summing over k yields
n − 1 ≥
�xi − xk�2 =
(diam A) 2
i=1
λi xi
n�
i=1
λi
x 2 i + x2 k − 2 xi · xk
(diam A) 2
� � 2σ
· xk =
2
for k = 1,...,n.
(diam A) 2
2nσ 2 �
n − 1
� 1 �
2 d
� 1
2
, or σ ≤ diam A ≤
diam A.
(diam A) 2 2n
2d + 2
This concludes the proof of (1).
For general A consider, for each point of A, the ball with centre at this point and
radius ϱ.By(1),anyd + 1 of these balls have non-empty intersection. Thus all these
balls have non-empty intersection by Helly’s theorem. Any ball with radius ϱ and
centre at a point of this intersection then contains each point of A. ⊓⊔