Attila Jung, 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 $F$ of $n$ convex sets in ${\mathbb{R}}^d$ such that at least $\alpha (\genfrac{}{}{0ex}{}{n}{d+1})$ of the $(d+1)$-tuples of $F$ have an intersection of volume at least 1, then one can select ${\mathrm{\Omega }}_{d,\alpha }(n)$ members of $F$ whose intersection has volume at least ${\mathrm{\Omega }}_d(1)$. Joint work with Nora Frankl and Istvan Tomon.