Proof of a Conjecture of Bárány, Katchalski and Pach

Abstract

Bárány, Katchalski and Pach (Proc Am Math Soc 86(1):109–114, 1982) (see also Bárány et al., Am Math Mon 91(6):362–365, 1984) proved the following quantitative form of Helly’s theorem. If the intersection of a family of convex sets in $\mathbb {R}^d$ is of volume one, then the intersection of some subfamily of at most 2d members is of volume at most some constant v(d). In Bárány et al. (Am Math Mon 91(6):362–365, 1984), the bound $v(d)\le d^{2d^2}$ was proved and $v(d)\le d^{cd}$ was conjectured. We confirm it.