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

Abstract

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 R d in general position, there exists a point of R d contained in at least 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. © 2012 Springer Science+Business Media, LLC.