Classifying unavoidable 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 $\{1,2,\dots ,T(d,r)\}$ into $r$ parts a "Tverberg type". We say that $\mathcal{I}$ "occurs" in an ordered point sequence $P$ if $P$ contains a subsequence $P^{\prime }$ of $T(d,r)$ points such that the partition of $P^{\prime }$ 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.

Revision following referee comments. 32 pages, 8 figures

References in corpus (1)

• A note on the tolerated Tverberg theorem

Cited by in corpus (1)

• On affine Tverberg-type results without continuous generalization