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 c such that for every set P of n points in R in general position, there exists a point of R 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 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 for arbitrary d. In particular, we improve the lower bound on c from 0.06332 to more than 0.07480; the best upper bound known on c being 0.09375. [ABSTRACT FROM AUTHOR]