# Regular Polygonal Tverberg-Type Theorems

## Bard College, New York

## Date and time: 2pm, Friday, March 22, 2019

## 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).