Quantitative Tverberg theorems over lattices and other discrete sets

TL;DR

This paper extends Tverberg's theorem to discrete sets in Euclidean space, establishing conditions for partitions with intersecting convex hulls containing multiple points, supported by new quantitative Helly and Carathéodory theorems.

Contribution

It introduces a novel variation of Tverberg's theorem applicable to discrete sets, with new quantitative versions of Helly's and Carathéodory's theorems to support the results.

Findings

Derived bounds on the number of points needed for Tverberg partitions.

Established quantitative Helly's theorem for discrete sets.

Proved quantitative Carathéodory's theorem for discrete sets.

Abstract

This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the main results require new quantitative versions of Helly's and Carath\'eodory's theorems.