On Gromov’s Method of Selecting Heavily Covered Points

A result of Boros and Füredi (d = 2) and of Bárány (arbitrary d) asserts that for every d there exists cd> 0 such that for every n-point set P ⊂ Rd, some point of Rd is covered by at least cd n d+1 of the d-simplices spanned by the points of P. The largest possible value of cd has been the subject of ongoing research. Recently Gromov improved the existing lower bounds considerably by introducing a new, topological proof method. We provide an exposition of the combinatorial component of Gromov’s approach, in terms accessible to combinatorialists and discrete geometers, and we investigate the limits of his method. In particular, we give tighter bounds on the cofilling profiles for the (n − 1)-simplex. These bounds yield a minor improvement over Gromov’s lower bounds on cd for large d, but they also show that the room for further improvement through the cofilling profiles alone is quite small. We also prove a slightly better lower bound for c3 by an approach using an additional structure besides the cofilling profiles. We formulate a combinatorial extremal problem whose solution might perhaps lead to a tight lower bound for cd