TATERS

Helly’s theorem from 1912 asserts that for a finite family of convex sets in a d-dimensional Euclidean space, if every d+1 of the sets have a point in common then all of the sets have a point in common. This theorem found applications in many areas of mathematics and led to numerous generalizations. Its proofs are very elementary and are suitable for undergraduate students and advanced high-school students.  

Helly’s theorem is closely related to two other fundamental theorems in convexity: Radon’s theorem asserts that a set of d + 2 points in d-dimensional real space can be divided into two disjoint sets whose convex hulls have non empty intersection. Caratheodory’s theorem asserts that if S is a set in d- dimensional real space and x belongs to its convex hull then x already belongs to the convex hull of at most d + 1 points in S.

We will discuss several developments around Helly’s theorem. 

One particular deep and mysterious theorem in this area is Tverberg's theorem that asserts that n points in d-dimensional Euclidean space with n = (d+1)(r-1)+1, can be divided into r parts whose convex hulls have non-empty intersection. 

If time allows we will discuss various quantitative versions of Helly’s theorem, "(p, q)"- theorems, and the beautiful Amenta’s theorem (a result about families of unions of convex sets). We will also mention some connections to topology.