Gruber P. Convex and Discrete Geometry

The Theorems of Radon, Carathéodory and Helly
3 Convex Sets, Convex Bodies and Convex Hulls 47
Above the theorem of Carathéodory was proved, following Radon. If the argument
used by Radon is formulated properly, it is called Radon’s theorem. The theorems of
Radon, Carathéodory and Helly, are equivalent in the sense that each can be proved
using any of the other ones. The equivalence is stated in several references, for
example in [284], p. 430, but we were not able to locate in the literature a complete
proof in the context of E d . Also here, we do not prove full equivalence.
Theorem 3.2. The following statements hold:
(i) Radon’s Theorem. Let A �= ∅be a set of at least d + 2 points in E d . Then
there are subsets B, C of A such that
B ∩ C =∅, conv B ∩ conv C �= ∅.
(ii) Carathéodory’s Theorem. Let A �= ∅be a set in E d . Then conv Aisthe
set of all convex combinations of affinely independent points in A.
(iii) Helly’s Theorem. Let F be a family of convex bodies in E d . If any d + 1 convex
bodies in F have non-empty intersection, then the intersection of all convex
bodies in F is non-empty.
Our proof of the implication (i)⇒(ii), in essence, is that of Carathéodory’s theorem
in Sect. 3.1. The proof of the implication (ii)⇒(iii) is due to Rademacher and
Schoenberg [819].
Proof. (i) It is sufficient to prove Radon’s theorem for A ={x1,...,xd+2} ⊆E d .
There are µ1,...,µd+2 ∈ R, not all 0, such that
µ1 +···+µd+2 = 0,
µ1x1 +···+µd+2xd+2 = 0.
We clearly may assume that µ1,...,µk ≥ 0 and −λk+1 = µk+1,...,−λd+2 =
µd+2 ≤ 0. Then
and thus
µ1 +···+µk = λk+1 +···+λd+2 > 0,
µ1x1 +···+µkxk = λk+1xk+1 +···+λd+2xd+2
µ1x1 +···+µkxk
µ1 +···+µk
= λk+1xk+1 +···+λd+2xd+2
.
λk+1 +···+λd+2
Now put B ={x1,...,xk}, C ={xk+1,...,xd+2}.
(i)⇒(ii) Let x ∈ conv A. By Lemma 3.1, x = λ1x1 + ··· + λnxn, where
λ1,...,λn > 0, λ1 + ··· + λn = 1, x1,...,xn ∈ A. We suppose that this representation
is chosen such that n is minimal. We have to show that x1,...,xn are
affinely independent.