Gruber P. Convex and Discrete Geometry

48 Convex Bodies
Suppose not. Then, by Radon’s theorem in aff {x1,...,xn}, there are disjoint
subsets of {x1,...,xn}, the convex hulls of which have non-empty intersection. By
re-numbering, if necessary, we thus may assume that
ν1x1 +···+νmxm = νm+1xm+1 +···+νnxn, or ν1x1 +···−νnxn = o,
where ν1,...,νn ≥ 0,ν1 +···+νm = νm+1 +···+νn = 1,
and thus ν1 +···−νn = 0.
Thus, up to notation, we have the same situation as in the proof of the Theorem 3.1
of Carathéodory which easily leads to a contradiction.
(ii)⇒(iii) Since the intersection of a family of compact sets is non-empty, if each
finite subfamily has non-empty intersection, it is sufficient to prove (iii) for finite
families F ={C1,...,Cn}, say, where n ≥ d + 2.
Assume that (iii) does not hold for F. Consider the function δ : E d → R
defined by
δ(x) = max{δ(x, Ci) : i = 1,...,n} for x ∈ E d ,
where δ(x, Ci) = min{�x − y� :y ∈ Ci}.
Let δ assume its minimum at p ∈ E d , say. Since (iii) does not hold for F, δ(p) >0.
By re-numbering, if necessary, we may suppose that
Choose qi ∈ Ci such that
Then
δ(p) = δ(p, Ci) precisely for i = 1,...,m (≤ n).
δ(p) = δ(p, Ci) =�p − qi� for i = 1,...,m.
p ∈ conv{q1,...,qm},
since otherwise we could decrease δ(p, Ci) and thus δ(p) by moving p closer to
conv{q1,...,qm}. By Carathéodory’s theorem there is a subset of {q1,...,qm} of
k ≤ d + 1 points, such that p is in the convex hull of this subset. By re-numbering,
if necessary, we may assume that this subset is the set {q1,...,qk}. Then
Clearly,
p = λ1q1 +···+λkqk, where λ1,...,λk ≥ 0, λ1 +···+λk = 1.
Ci ⊆ H −
i = � x : (x − p) · (qi − p) ≥�qi − p� 2 (> 0) � for i = 1,...,k.
Since k ≤ d + 1, we may choose a point y ∈ C1 ∩···∩Ck ⊆ H −
1 ∩···∩H − k . Then
0 = (y − p) · (p − p) = (y − p) · (λ1q1 +···+λkqk − p)
= (y − p) · � λ1(q1 − p) +···+λk(qk − p) �
= λ1(y − p) · (q1 − p) +···+λk(y − p) · (qk − p) >0,
which is the desired contradiction. ⊓⊔