Tverberg Type Partitions: Sub-Regular and Elliptical Polygons

Author

Tobias Golz Timofeyev, Bard CollegeFollow

Date of Submission

Spring 2021

Academic Program

Mathematics

Project Advisor 1

Steven Simon

Abstract/Artist's Statement

Tverberg's theorem states that given a set S of T(r,d)=(r-1)(d+1)+1 points in R^d, there exists a partition of S into r subsets whose convex hulls intersect. A feature of Tverberg's theorem is that T(r,d) is tight, so in this senior project we investigate Tverberg-type results when |S|. We found that in R², given a set S of T(r,2)-2=3r-4 points, and assuming r=r₁ r₂, there exists a partition of S into r sets such that when grouped into r₁ collections of r₂ sets, the convex hulls of each collection overlap, and we can find the vertex set of a regular r₁-gon with one point from the intersection of each collection. We also show that given a similar construction but with |S|=3r-6, we can find the vertices of an r₁-gon in the intersections of convex hulls, with vertices on an ellipse, and other nice regularity properties.