Helly type theorems for open connected sets in the plane

Seminar
Speaker
Minki Kim (Technion)
Date
11/11/2018 - 15:30 - 14:00Add to Calendar 2018-11-11 14:00:00 2018-11-11 15:30:00 Helly type theorems for open connected sets in the plane Helly's theorem is a classical result in combinatorial geometry about the intersection patterns of convex sets in Euclidean spaces. For the 2-dimensional case, it asserts that for a family of convex sets in the plane, if every 3 members have a point in common, then all members in the family have a point in common. The topological Helly theorem, also proved by Helly, shows that the same assertion holds for a family of open and contractible sets where each intersection is again open and contractible. In this talk, I will present Helly-type theorems for a more general set-system: a family of open connected sets in the plane where any intersection is again open and connected. In this setting, if every 4 members have a point in common, then all members in the family have a point in common. I will also discuss a fractional generalization of this. Room 201, Math and CS Building (Bldg. 216) אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Room 201, Math and CS Building (Bldg. 216)
Abstract

Helly's theorem is a classical result in combinatorial geometry about the intersection patterns of convex sets in Euclidean spaces. For the 2-dimensional case, it asserts that for a family of convex sets in the plane, if every 3 members have a point in common, then all members in the family have a point in common. The topological Helly theorem, also proved by Helly, shows that the same assertion holds for a family of open and contractible sets where each intersection is again open and contractible.

In this talk, I will present Helly-type theorems for a more general set-system: a family of open connected sets in the plane where any intersection is again open and connected. In this setting, if every 4 members have a point in common, then all members in the family have a point in common. I will also discuss a fractional generalization of this.

Last Updated Date : 06/11/2018