Abstract
A result of Boros and Furedi (d=2) and of Barany (arbotrary d) asserts that for every d there exists a constant c_d such that for every point set P in R^d, some point of P is covered by at least the constant c_d-portion of all of the d-simplices spanned by the points of P. The largest possible value of c_d 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 c_d 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 c_3 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 c_d.