Proof of a Conjecture of Barany, Katchalski and Pach

Barany, Katchalski and Pach (Proc Am Math Soc 86(1):109-114, 1982) (see also Barany 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 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 Barany et al. (Am Math Mon 91(6):362-365, 1984), the bound was proved and was conjectured. We confirm it.