# The Crossing Tverberg Theorem

## TL;DR

This paper strengthens Tverberg's theorem by ensuring partitions with convex hulls that have pairwise intersecting boundaries, leading to new geometric configurations such as crossing triangles in the plane, with implications for higher dimensions.

## Contribution

It introduces a strengthened version of Tverberg's theorem that guarantees pairwise intersecting convex hull boundaries, extending geometric intersection results to higher dimensions.

## Key findings

- Proves a strengthened Tverberg theorem with intersecting boundaries
- Shows any n points in the plane span approximately n/3 crossing triangles
- Generalizes the crossing property to simplices in higher dimensions

## Abstract

Tverberg's theorem is one of the cornerstones of discrete geometry. It states that, given a set $X$ of at least $(d+1)(r-1)+1$ points in $\mathbb R^d$, one can find a partition $X=X_1\cup \ldots \cup X_r$ of $X$, such that the convex hulls of the $X_i$, $i=1,\ldots,r$, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any $n$ points in the plane in general position span $\lfloor n/3\rfloor$ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Rebollar et al.\ guarantees $\lfloor n/6\rfloor$ pairwise crossing triangles. Our result generalizes to a result about simplices in $\mathbb R^d,d\ge2$.