Mathematics Colloquia and Seminars

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The study of the intersection properties of convex sets is a central topic in discrete geometry. One of the most important results in this area is Tverberg's theorem, proved in 1966. This theorem guarantees that for any set of points $S\subset \mathbb{R}^d$ sufficiently large, there exists a partition into $r$ subsets whose convex hulls intersect, such partitions are called Tverberg partitions.
Not much is known about the structure of such partitions. Even finding optimal lower bounds for the number of Tverberg partitions when $|S| =(d + 1)(r-1) + $1 is a famous open problem known as the Sierksma's conjecture.

In this talk, we will talk about the Tverberg partition graph of a set of points $S$, which has the set of Tverberg partitions as vertices and where the edge connectivity measures how different the partitions are. We will analyze some of the structural properties of said graph, such as: the connectivity, the clan number and the distribution of the degrees.