Tverberg theorems over discrete sets of points

Abstract

This paper discusses Tverberg-type theorems with coordinate constraints (i.e., versions of these theorems where all points lie within a subset $S\subset {\mathbb{R}}^d$ and the intersection of convex hulls is required to have a non-empty intersection with $S$). We determine the $m$-Tverberg number, when $m\ge 3$, of any discrete subset $S$ of ${\mathbb{R}}^2$ (a generalization of an unpublished result of J.-P. Doignon). We also present improvements on the upper bounds for the Tverberg numbers of ${\mathbb{Z}}^3$ and ${\mathbb{Z}}^j\times {\mathbb{R}}^k$ and an integer version of the well-known positive-fraction selection lemma of J. Pach.

14 pages, 1 figure