Inscribed Tverberg-Type Partitions for Orbit Polytopes

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.