Inscribed Tverberg-Type Partitions for Orbit Polytopes

TL;DR

This paper extends Tverberg's theorem by demonstrating that for certain symmetry conditions and group actions, generic point sets can be partitioned into subsets whose convex hulls contain vertices of a convex polytope with a specified isometry group.

Contribution

It introduces a method to guarantee inscribed polytopal partitions with symmetry constraints for generic point sets, generalizing Tverberg's theorem to include group actions and isometry groups.

Findings

Partitions exist for all regular polygons in the plane.

Partitions exist for some regular 4-polytopes in 4D.

Number of points used is optimal, matching Tverberg's bounds.

Abstract

Tverberg's theorem 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. Moreover, generic collections of fewer points cannot be so divided. Extending earlier work of the first author, we show that one can nonetheless guarantee inscribed ``polytopal partitions" with specified symmetry conditions in many such circumstances. Namely, for any faithful and full--dimensional orthogonal representation $\rho\colon G\rightarrow O(d)$ of any order $r$ group $G$, we show that a generic set of $t(r,d)-d$ points in $\mathbb{R}^d$ can be partitioned into $r$ subsets so that there are $r$ points, one from each of the resulting convex hulls, which are the vertices of a convex $d$--polytope whose isometry group contains $G$ via the regular action afforded by the representation. As with Tverberg's theorem, the number of points is optimal for this. At one extreme, this gives polytopal partitions for all regular $r$--gons in the plane, as well as for three of the six regular 4--polytopes in $\mathbb{R}^4$. At the other extreme, one has polytopal partitions for $d$-polytopes on $r$ vertices with isometry group equal to $G$ whenever $G$ is the isometry group of a vertex--transitive $d$-polytope.