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 Rd in general position, there exists a point of Rd contained in at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{d}\binom{n}{d+1}$\end{document}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.