Tverberg partitions as weak epsilon-nets

TL;DR

This paper introduces a probabilistic approach to Tverberg-type theorems, establishing minimal partition counts to guarantee Tverberg partitions on large subsets, thus extending known tolerance results.

Contribution

It provides a new probabilistic proof and generalizes Tverberg's theorem with tolerance by determining minimal partitions for large subsets.

Findings

Established probabilistic method for Tverberg partitions

Derived minimal partition counts for subsets with size proportional to total

Extended known results about Tverberg's theorem with tolerance

Abstract

We prove a Tverberg-type theorem using the probabilistic method. Given $\varepsilon >0$, we find the smallest number of partitions of a set $X$ in $R^d$ into $r$ parts needed in order to induce at least one Tverberg partition on every subset of $X$ with at least $\varepsilon |X|$ elements. This generalizes known results about Tverberg's theorem with tolerance.