Ugrás a tartalomhoz

## Convex Geometry

Csaba Vincze (2013)

University of Debrecen

3. fejezet - Helly's theorem

## 3. fejezet - Helly's theorem

Helly's theorem gives a criteria to provide the existence of common elements in each member of a family of convex sets in the space. The one-dimensional version is that if we have a finite collection of intervals and any twoof them have a common point then all of them have a common point. For an alternative formulation image that each interval represents the time that a guest spends at a party. The existence of the common point of each pair of the intervals corresponds to the moment when two guests welcome to each other. It is clear that if x denotes the guest who is the first to leave the party at the moment t(0) then there is no any guest who arrives after t(0) otherwise such a guest can not welcome to x. On the other hand there is no any guest to leave the party before t(0) because x is the first. Therefore t(0) is a moment when all the guests are at the party at.

Lemma 3.1.1 (Radon, Johann). Let D be the set consisting of the elements

 ${v}_{1},\mathrm{\dots },{v}_{k}$

in the coordinate space of dimension n. If k is at least n+2 then D can be partitioned into two disjoint subsets such that their convex hulls intersect each other, i.e.

but

 $conv{D}_{1}\cap conv{D}_{2}\ne \mathrm{\varnothing }.$

Proof Since k is at least n+2 the elements in D are affinely dependent, i.e. we have a non-trivial k-tuple of scalar multipliers such that

 ${\lambda }_{1}{v}_{1}+\mathrm{\dots }+{\lambda }_{k}{v}_{k}=0$ (3.1)

and λ(1)+ ...+λ(k)=0. Because the sum of the coefficients is zero there must be numbers with different signs among them. For the sake of definiteness suppose that

Let

 $\lambda :={\lambda }_{1}+\mathrm{\dots }+{\lambda }_{l}=-\left({\lambda }_{l+1}+\mathrm{\dots }+{\lambda }_{k}\right)>0$

Then, by 3.1

 $v:=\frac{1}{\lambda }\left({\lambda }_{1}{v}_{1}+\mathrm{\dots }+{\lambda }_{k}{v}_{k}\right)=-\frac{1}{\lambda }\left({\lambda }_{l+1}{v}_{l+1}+\mathrm{\dots }{\lambda }_{k}{v}_{k}\right)$

and the element v is contained in the convex hulls of both

as was to be proved. ▮