Given a probability measure on $R^d$, what is the probability that the
simplex spanned by $d+1$ randomly selected points contains the origin
in its interior? To study this question and its relatives, we
introduce the following definition. The overlap number of a finite
$(d+1)$-uniform hypergraph $H$ is the largest constant $c(H)\in (0,1]$
such that no matter how we map the vertices of $H$ into $R^d$, there
is a point covered by at least a $c(H)$-fraction of the simplices
induced by the images of its hyperedges. We survey some old and new
results related to this concept. Joint work with J. Fox, M. Gromov, V.
Lafforgue, and A. Naor.

[Invited by Prof. Ruth Kellerhals]