Csaba Vincze (2013)

University of Debrecen

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.

$D={D}_{1}\cup {D}_{2}\text{}and\text{}{D}_{1}\cap {D}_{2}=\mathrm{\varnothing}$ |

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

${\lambda}_{1}\ge 0,\mathrm{\dots},{\lambda}_{l}\ge 0\text{}and\text{}{\lambda}_{l+1}0,\dots ,{\lambda}_{k}0.$ |

Let

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

Then, by 3.1

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

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

${D}_{1}:=\{{v}_{1},\mathrm{\dots},{v}_{l}\}\text{}and\text{}{D}_{2}:=\{{v}_{l+1},\mathrm{\dots},{v}_{k}\}$ |

as was to be proved. ▮