3:00–4:00 pm
Eckhart Hall Eckhart Hall, Room 202 5734 S. University Ave. Chicago, IL 60637
Helly Type Problems
Abstract:
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. I will mention problems around Helly's theorem with combinatorial, topological and geometrical flavours.