Tverberg plus minus

TL;DR

This paper proves a generalized Tverberg theorem allowing a specified number of negative coefficients in the affine combination representing the intersection point, extending classical Tverberg results.

Contribution

It introduces a new Tverberg-type theorem that characterizes partitions with a controlled number of negative coefficients in the affine combination.

Findings

Established a new partition theorem with negative coefficients.

Extended classical Tverberg theorem to include negative affine coefficients.

Provided conditions for the existence of such partitions in general position sets.

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 $k\in \{0,1,\ldots,r-1\}$, there is a partition of $A$ into $r$ sets $A_1,\ldots,A_r$ 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.