  We propose a new representation of $k$-partite, $k$-uniform hypergraphs, that
is, a hypergraph with a partition of vertices into $k$ parts such that each
hyperedge contains exactly one vertex of each type; we call them
$k$-hypergraphs for short. Given positive integers $\ell, d$, and $k$ with
$\ell\leq d-1$ and $k={d\choose\ell}$, any finite set $P$ of points in
$\mathbb{R}^d$ represents a $k$-hypergraph $G_P$ as follows. Each point in $P$
is covered by $k$ many axis-aligned affine $\ell$-dimensional subspaces of
$\mathbb{R}^d$, which we call $\ell$-subspaces for brevity and which form the
vertex set of $G_P$. We interpret each point in $P$ as a hyperedge of $G_P$
that contains each of the covering $\ell$-subspaces as a vertex. The class of
\emph{$(d,\ell)$-hypergraphs} is the class of $k$-hypergraphs that can be
represented in this way. The resulting classes of hypergraphs are fairly rich:
Every $k$-hypergraph is a $(k,k-1)$-hypergraph. On the other hand,
$(d,\ell)$-hypergraphs form a proper subclass of the class of all
$k$-hypergraphs for $\ell<d-1$.
  In this paper we give a natural structural characterization of
$(d,\ell)$-hypergraphs based on vertex cuts. This characterization leads to a
poly\-nomial-time recognition algorithm that decides for a given $k$-hypergraph
whether or not it is a $(d,\ell)$-hypergraph and that computes a representation
if existing. We assume that the dimension $d$ is constant and that the
partitioning of the vertex set is prescribed.
