On June 17, 2025, Attila Jung from the Eötvös Loránd University, Hungary gave a talk at the Discrete Math Seminar on combining the fractional Helly theorem and the quantitative volume theorem on convex sets optimally. The title of his talk was “The Quantitative Fractional Helly Theorem“.
Two celebrated extensions of Helly’s theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of Barany, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combination of these two results. We show that given a family of convex sets in such that at least of the -tuples of have an intersection of volume at least 1, then one can select members of whose intersection has volume at least . Joint work with Nora Frankl and Istvan Tomon.
