Here is an abstract for the talk decorated (it may take me a few days) with links to some earlier posts where matters are described in details, and also with some external links.
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. 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.
The first part of the lecture will discuss the basic relation between Helly, Radon, and Caratheodory’s theorems. Those are described in this post We move to discuss several quantitative generalizations of the theorem. At first we will consider what happens to the theorem if we replace the condition “non-empty intersection” by “the dimension of the intersection is at least r. (Question: What is the answer for r=1?)
Meir Katchalski settled this problem for every d and r in his M. Sc thesis. Later he proved the “Hard-Katchalski theorem” that asserts that if we know the dimensions of intersection for every subfamily of cardinality at most d+1, we can figure out the dimensions of intersection for every subfamily.
After that we will discuss various quantitative versions of Helly’s theorem. The upper bound theorem for families of convex sets (Katchalski-Perles conjecture), fractional Helly’s theorems, and results about the volume of intersections.
Our next topic will be Tverberg’s theorem (a far-reaching extension of Radon’s theorem) and we will indicate the proof. Next we describe the beautiful Amenta’s theorem – a Helly-type result about families of unions of convex sets. We will briefly mention Alon-Kleitman (p,q) theorem, (If from every p sets q have a point in common what can you deduce?) We conclude with the topological Helly’s theorem discovered by Helly himself and discuss some topological versions of the results considered above. (Here we replace convex sets by topological balls and assume that also that nonempty intersections are topological balls.)