  The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is
defined as 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.
In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph
expansion for higher dimensional simplicial complexes, it was asked whether or
not there exists a sequence $\{H_n\}_{n=1}^\infty$ of arbitrarily large
$(d+1)$-uniform hypergraphs with bounded degree, for which $\inf_{n\ge 1}
c(H_n)>0$. Using both random methods and explicit constructions, we answer this
question positively by constructing infinite families of $(d+1)$-uniform
hypergraphs with bounded degree such that their overlap numbers are bounded
from below by a positive constant $c=c(d)$. We also show that, for every $d$,
the best value of the constant $c=c(d)$ that can be achieved by such a
construction is asymptotically equal to the limit of the overlap numbers of the
complete $(d+1)$-uniform hypergraphs with $n$ vertices, as
$n\rightarrow\infty$. For the proof of the latter statement, we establish the
following geometric partitioning result of independent interest. For any $d$
and any $\epsilon>0$, there exists $K=K(\epsilon,d)\ge d+1$ satisfying the
following condition. For any $k\ge K$, for any point $q \in \mathbb{R}^d$ and
for any finite Borel measure $\mu$ on $\mathbb{R}^d$ with respect to which
every hyperplane has measure $0$, there is a partition $\mathbb{R}^d=A_1 \cup
\ldots \cup A_{k}$ into $k$ measurable parts of equal measure such that all but
at most an $\epsilon$-fraction of the $(d+1)$-tuples
$A_{i_1},\ldots,A_{i_{d+1}}$ have the property that either all simplices with
one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain
$q$.
