Tverberg's theorem, a new proof

Abstract

We give a new proof Tverberg's famous theorem: For every set $X\subset {\mathbb{R}}^d$ with $|X|=(r-1)(d+1)+1$, there is a partition of $X$ into $r$ sets $X_1,\dots ,X_r$ 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.