BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Zuzana Patáková (Charles University) DTSTART:20210520T120000Z DTEND:20210520T130000Z DTSTAMP:20260423T040114Z UID:DCGParis/6 DESCRIPTION:Title: On Radon and fractional Helly theorems\nby Zuzana Patáková (Charles University) as part of Discrete and Computational Geometry Seminar in Par is\n\n\nAbstract\nRadon theorem plays a basic role in many results of comb inatorial convexity. It says that any set of d+2 points in R^d can be spli t into two parts so that their convex hulls intersect. It implies Helly th eorem and as shown recently also its more robust version\, so-called fract ional Helly theorem. By standard techniques this consequently yields an ex istence of weak epsilon nets and a (p\,q)-theorem.\n\nWe will show that we can obtain these results even without assuming convexity\, replacing it w ith very weak topological conditions. More precisely\, given an intersecti on-closed family F of subsets of R^d\, we will measure the complexity of F by the supremum of the first d/2 Betti numbers over all elements of F. We show that bounded complexity of F guarantees versions of all the results mentioned above.\n\nBased on joint work with Xavier Goaoc and Andreas Holm sen.\n LOCATION:https://researchseminars.org/talk/DCGParis/6/ END:VEVENT END:VCALENDAR