On Gromov's Method of Selecting Heavily Covered Points

A result of Boros and F\"uredi ($d=2$) and of B\'ar\'any (arbitrary $d$) asserts that for every $d$ there exists $c_d>0$ such that for every $n$-point set $P\subset \R^d$, some point of $\R^d$ is covered by at least $c_d{n\choose d+1}$ 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 \emph{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 $c_d$.