# Regular Polygonal Tverberg-Type Theorems

## Place: CUNY Graduate Center, Rm 4419

Tverberg's Theorem states that any $T(q,d)=(q-1)(d+1) +1$ points in $\mathbb{R}^d$ can be partitioned into $q$ pairwise disjoint sets whose convex hulls have non-empty $q$-fold intersection. This number $T(q,d)$ is generically tight. For fewer than $T(q,d)$ points, we will show that in lieu of a full Tverberg partition one can still guarantee a partition into $q$ pairwise disjoint sets so that there are $q$ points, one from each of the resulting convex hulls, which form the vertices of a regular $q$-gon. Analogous results hold for prismatic polytopes. As with Tverberg's theorem, these results have continuous extensions when $q$ is a prime power, where they admit constrained versions (e.g., restrictions on the dimension of the convex hulls, or requiring that the vertices are equidistant to the original point set).