Tverberg's theorem, a new proof

TL;DR

This paper presents a new proof of Tverberg's theorem using linear algebra and structured matrices, offering an alternative approach to the original geometric proof.

Contribution

The paper introduces a novel proof of Tverberg's theorem leveraging linear algebra techniques and structured matrices, differing from traditional geometric methods.

Findings

New proof confirms Tverberg's theorem

Utilizes linear algebra and structured matrices

Provides alternative perspective on the theorem

Abstract

We give a new proof Tverberg's famous theorem: For every set $X \subset \R^d$ with $|X|=(r-1)(d+1)+1$, there is a partition of $X$ into $r$ sets $X_1,\ldots,X_r$ such that $\bigcap_{p=1}^r \conv X_p\ne \emptyset$. The new proof uses linear algebra, specially structured matrices, the theory of linear equations, and Tverberg's original ``moving the points" method.