On Tverberg partitions

TL;DR

This paper explores the structure of Tverberg partitions, showing that for certain partitions of the point set size, all Tverberg partitions induce the same partition of the set, revealing a new structural property.

Contribution

It proves that for any partition with parts at most d+1, all Tverberg partitions of a constructed set induce the same partition of the point set.

Findings

Existence of sets with uniform Tverberg partition structure

All Tverberg partitions induce the same partition for certain size constraints

Provides insight into the combinatorial structure of Tverberg partitions

Abstract

A theorem of Tverberg from 1966 asserts that every set $X\subset\mathbb{R}^d$ of $n=T(d,r)=(d+1)(r-1)+1$ points can be partitioned into $r$ pairwise disjoint subsets, whose convex hulls have a point in common. Thus every such partition induces an integer partition of $n$ into $r$ parts (that is, $r$ integers $a_1,\ldots,a_r$ satisfying $n=a_1+\cdots+a_r$), where the parts $a_i$ correspond to the number of points in every subset. In this paper, we prove that for any partition $a_i\le d+1$, $i=1,\ldots,r$, there exists a set $X\subset\mathbb{R}^d$ of $n$ points, such that every Tverberg partition of $X$ induces the same partition on $n$, given by the parts $a_1,\ldots,a_r$.