Summary: Boros and Füredi (for \(d=2\)) and Bárány (for arbitrary \(d\)) proved that there exists a positive real number \(c_d\) such that for every set \(P\) of \(n\) points in \({\mathbf R}^d\) in general position, there exists a point of \({\mathbf R}^d\) contained in at least \(c_d \binom{n}{d+1}\) \(d\)-simplices with vertices at the points of \(P\). Gromov improved the known lower bound on \(c_d\) by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov’s approach and thereby provide a new stronger lower bound on \(c_d\) for arbitrary \(d\). In particular, we improve the lower bound on \(c_3\) from \(0.06332\) to more than \(0.07480\); the best upper bound known on \(c_3\) being \(0.09375\).