  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.
