A New Lower Bound Based on Gromov's Method of Selecting Heavily Covered Points

Boros and Füredi (for d = 2) and Bárány (for arbitrary d) proved that there exists a positive real number cd such that for every set P of n points in R d in general position, there exists a point of R contained in at least cd ( n d+1 ) d-simplices with vertices at the points of P . Gromov improved the known lower bound on cd 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 cd for arbitrary d. In particular, we improve the lower bound on c3 from 0.06332 to more than 0.07480; the best upper bound known on c3 being 0.09375.