MIT-Harvard-MSR Combinatorics Seminar: New quantitative Helly-type theorems for diameter

Helly’s theorem is a fundamental statement in discrete and convex geometry  that relates the intersection of a family of convex sets to the  intersections of its subfamilies. This talk surveys recent advances in  quantitative versions of Helly’s theorem, including best-known results  toward proving a 1982 conjecture of Bárány, Katchalski, and Pach. Along  the way, I’ll introduce a new, surprisingly powerful technique for proving  quantitative Helly-type theorems, and we’ll completely characterize the  norms for which there is a “no-loss” Helly-type theorem for diameter.