Regular Polygonal Partitions of a Tverberg Type

TL;DR

This paper extends Tverberg's theorem by demonstrating that certain point sets in Euclidean space can be partitioned into subsets whose convex hulls intersect in a pattern matching regular polygons or their products, even with fewer points than the classical theorem requires.

Contribution

It introduces new partitioning results for point sets related to regular polygons and their products, generalizing Tverberg's theorem to these geometric configurations and establishing tight bounds.

Findings

Partitioning of points into subsets with convex hulls intersecting as regular polygons

Extension of Tverberg's theorem to polygonal and product polytopes

Topological and dimensionally restricted generalizations

Abstract

A seminal theorem of Tverberg states that any set of $T(r,d)=(r-1)(d+1)+1$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets whose convex hulls have non-empty $r$-fold intersection. Almost any collection of fewer points in $\mathbb{R}^d$ cannot be so divided, and in these cases we ask if the set can nonetheless be $P(r,d)$--partitioned, i.e., split into $r$ subsets so that there exist $r$ points, one from each resulting convex hull, which form the vertex set of a prescribed convex $d$--polytope $P(r,d)$. Our main theorem shows that this is the case for any generic $T(r,2)-2$ points in the plane and any $r\geq 3$ when $P(r,2)=P_r$ is a regular $r$--gon, and moreover that $T(r,2)-2$ is tight. For higher dimensional polytopes and $r=r_1\cdots r_k$, $r_i \geq 3$, this generalizes to $T(r,2k)-2k$ generic points in $\mathbb{R}^{2k}$ and orthogonal products $P(r,2k)=P_{r_1}\times \cdots \times P_{r_k}$ of regular polygons, and likewise to $T(2r,2k+1)-(2k+1)$ points in $\mathbb{R}^{2k+1}$ and the product polytopes $P(2r,2k+1)=P_{r_1}\times \cdots \times P_{r_k} \times P_2$. As with Tverberg's original theorem, our results admit topological generalizations when $r$ is a prime power, and, using the "constraint method" of Blagojevi\'c, Frick, and Ziegler, allow for dimensionally restricted versions of a van Kampen--Flores type and colored analogues in the fashion of Sober\'on.