Tverberg theorems over discrete sets of points

TL;DR

This paper extends Tverberg theorems to discrete point sets in Euclidean spaces, providing exact and improved bounds for Tverberg numbers in various discrete and mixed spaces, with applications to selection lemmas.

Contribution

It determines the m-Tverberg number for any discrete subset of b2 and improves bounds for Tverberg numbers in higher-dimensional integer and mixed spaces.

Findings

Exact m-Tverberg number for discrete subsets of b2.

Improved upper bounds for Tverberg numbers in b3 and b j d7 b0k.

An integer version of the positive-fraction selection lemma.

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the $m$-Tverberg number, when $m \geq 3$, of any discrete subset $S$ of $\mathbb{R}^2$ (a generalization of an unpublished result of J.-P. Doignon).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.