Quentin Dubroff, Rutgers University

Abstract

Helly's theorem states that if any d+1 or fewer elements of a finite family of convex sets in R^d have non-empty intersection then there is a point which is contained in every member of the family. I will present a proof of this theorem and will discuss its foundational role in convex geometry. I will then show how we can extract information about intersection patterns of convex sets by studying simplicial complexes following a paper of Gil Kalai.