Motivated exposition of the proof of the Tverberg Theorem

TL;DR

This paper provides an accessible, motivated explanation of the proof of Tverberg's Theorem, emphasizing natural constructions and reducing complexity compared to traditional presentations.

Contribution

It offers a clearer, more intuitive exposition of Tverberg's Theorem proof by avoiding non-motivated preliminary constructions and highlighting natural geometric ideas.

Findings

Simplified proof presentation enhances understanding.

Natural geometric constructions replace complex preliminary steps.

The approach clarifies the reduction to Baryáni's Theorem.

Abstract

We present a motivated exposition of the proof of the following Tverberg Theorem: For every integers $d,r$ any $(d+1)(r-1)+1$ points in $\mathbb R^d$ can be decomposed into $r$ groups such that all the $r$ convex hulls of the groups have a common point. The proof is by well-known reduction to the B\'ar\'any Theorem. However, our exposition is easier to grasp because additional constructions (of an embedding $\mathbb R^d\subset\mathbb R^{d+1}$, of vectors $\varphi_{j,i}$ and statement of the Bara\'ny Theorem) are not introduced in advance in a non-motivated way, but naturally appear in an attempt to construct the required decomposition. This attempt is based on rewriting several equalities between vectors as one equality between vectors of higher dimension.