Regular Polygonal Partitions of a Tverberg Type

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.