Quantitative Helly-type theorems

Seminar
Speaker
Chaya Keller (Ben-Gurion University)
Date
05/11/2017 - 15:30 - 14:00Add to Calendar 2017-11-05 14:00:00 2017-11-05 15:30:00 Quantitative Helly-type theorems The classical Helly's theorem, dated 1923, asserts that if F is a family of compact convex sets in R^d such that any d+1 sets of F have a non-empty intersection, then all sets of F can be pierced by a single point. Helly's theorem is a cornerstone in convexity theory, and the need to generalize and extend it has led mathematicians to deep and fascinating new research directions. One of the best-known extensions is the Alon-Kleitman (p,q) theorem (1992) which asserts that for F as above, if among any p sets of F some q intersect (for q>d), then all sets of F can be pierced by a bounded number c(p,q,d) of points. In this talk we survey the quest for quantitative Helly-type theorems which aim at finding effective bounds on the piercing numbers in various scenarios. We present new bounds on c(p,q,d) for the Alon-Kleitman theorem, which are almost tight for a wide range of parameters. We also show that for several large classes of families, quantitative (p,2) theorems in the plane can be obtained, providing a strong connection between the piercing number of a family to its well-studied packing number, and giving rise to new Ramsey-type theorems. Based on joint works with Shakhar Smorodinsky and Gabor Tardos.  Room 201 , Bldg 216 - Math and CS Building אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Room 201 , Bldg 216 - Math and CS Building
Abstract

The classical Helly's theorem, dated 1923, asserts that if F is a family of compact convex sets in R^d such that any d+1 sets of F have a non-empty intersection, then all sets of F can be pierced by a single point. Helly's theorem is a cornerstone in convexity theory, and the need to generalize and extend it has led mathematicians to deep and fascinating new research directions. One of the best-known extensions is the Alon-Kleitman (p,q) theorem (1992) which asserts that for F as above, if among any p sets of F some q intersect (for q>d), then all sets of F can be pierced by a bounded number c(p,q,d) of points.

In this talk we survey the quest for quantitative Helly-type theorems which aim at finding effective bounds on the piercing numbers in various scenarios. We present new bounds on c(p,q,d) for the Alon-Kleitman theorem, which are almost tight for a wide range of parameters. We also show that for several large classes of families, quantitative (p,2) theorems in the plane can be obtained, providing a strong connection between the piercing number of a family to its well-studied packing number, and giving rise to new Ramsey-type theorems.

Based on joint works with Shakhar Smorodinsky and Gabor Tardos. 

Last Updated Date : 26/11/2019