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Tverberg's theorem states that given a set S of T(r,d)=(r-1)(d+1)+1 points in Rd, 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.
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Timofeyev, Tobias Golz, "Tverberg Type Partitions: Sub-Regular and Elliptical Polygons" (2021). Senior Projects Spring 2021. 249.
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