Tverberg plus minus

Abstract

We prove a Tverberg type theorem: Given a set $A\subset {\mathbb{R}}^d$ in general position with $|A|=(r-1)(d+1)+1$ and $k\in \{0,1,\dots ,r-1\}$, there is a partition of $A$ into $r$ sets $A_1,\dots ,A_r$ with the following property. The unique $z\in {\bigcap }_1^r\mathrm{aff}{A}_j$ can be written as an affine combination of the element in $A_j$: $z={\sum }_{x\in A_j}\alpha (x)x$ for every $p$ and exactly $k$ of the coefficients $\alpha (x)$ are negative. The case $k=0$ is Tverberg's classical theorem.