An elementary proof of Sierksma's conjecture for seven points in the plane

TL;DR

This paper presents a straightforward geometric proof confirming Sierksma's conjecture for seven points in the plane, establishing the existence of four Tverberg partitions into three sets.

Contribution

It provides the first elementary geometric proof for the seven-point case of Sierksma's conjecture, replacing previous topological methods.

Findings

Confirmed Sierksma's conjecture for seven points with four Tverberg partitions

Introduced a new simple geometric proof method

Established the only verified non-trivial case of the conjecture

Abstract

We give a new simple geometric proof that any seven points in the plane have four Tverberg partitions into three sets. This is the only confirmed non-trivial case of Sierksma's conjecture. Earlier proofs, by Stephan Hell, relied on topological arguments.