  B\'ar\'any, 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.
