Classifying unavoidable Tverberg partitions

TL;DR

This paper investigates which Tverberg partitions are unavoidable in long point sequences, proposes a conjecture for their complete characterization, and provides partial proofs and related results supporting Sierksma's conjecture.

Contribution

It introduces the concept of unavoidable Tverberg types, conjectures a full characterization, and proves some cases for dimensions up to 4, advancing understanding of Tverberg partitions.

Findings

Identified conditions under which Tverberg types are unavoidable.

Proved partial results for dimensions up to 4.

Constructed point sets with exactly $(r-1)!^d$ Tverberg partitions.

Abstract

Let $T(d,r) = (r-1)(d+1)+1$ be the parameter in Tverberg's theorem, and call a partition $\mathcal I$ of $\{1,2,\ldots,T(d,r)\}$ into $r$ parts a "Tverberg type". We say that $\mathcal I$ "occurs" in an ordered point sequence $P$ if $P$ contains a subsequence $P'$ of $T(d,r)$ points such that the partition of $P'$ that is order-isomorphic to $\mathcal I$ is a Tverberg partition. We say that $\mathcal I$ is "unavoidable" if it occurs in every sufficiently long point sequence. In this paper we study the problem of determining which Tverberg types are unavoidable. We conjecture a complete characterization of the unavoidable Tverberg types, and we prove some cases of our conjecture for $d\le 4$. Along the way, we study the avoidability of many other geometric predicates. Our techniques also yield a large family of $T(d,r)$-point sets for which the number of Tverberg partitions is exactly $(r-1)!^d$. This lends further support for Sierksma's conjecture on the number of Tverberg partitions.