  Boros and Furedi (for d=2) and Barany (for abritrary 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 c_d
n!/(d+1)!(n-d-1)! d-simplices with vertices at the points of P. Gromov improved
the 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.
