How do 9 points look like in $E^3$?

TL;DR

This paper provides an elementary proof of a geometric theorem in three-dimensional space, showing that for two groups of convex sets with mutual intersections, a line transversal exists for at least one group.

Contribution

It offers a simplified proof of a special case of a known theorem, extending the colorful Helly theorem to a new setting in $ ext{E}^3$.

Findings

Existence of a line transversal for at least one color group under given conditions

Extension of the colorful Helly theorem in three-dimensional space

Elementary proof technique for a geometric intersection property

Abstract

The aim of this note is to give an elementary proof of the following fact: given 3 red convex sets and 3 blue convex sets in $\mathbb{E}^3$, such that every red intersects every blue, there is a line transversal to the reds or there is a line transversal to the blues. This is a special case of a theorem of Montajano and Karasev \cite{MK} and generalizes, in a sense, the colourful Helly theorem due to Lov\'asz (cf. \cite{BL}).