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Proof of a conjecture of Bárány, Katchalski and Pach

Bárány, Katchalski and Pach 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 $v(d)$. They proved the bound $v(d)\leq d^{2d^2}$, and conjectured $v(d)\leq d^{cd}$. We confirm it.