  We study the problem of recognizing whether a given abstract simplicial
complex $K$ is the $k$-skeleton of the nerve of $j$-dimensional convex sets in
$\mathbb{R}^d$. We denote this problem by $R(k,j,d)$. As a main contribution,
we unify the results of many previous works under this framework and show that
many of these works in fact imply stronger results than explicitly stated. This
allows us to settle the complexity status of $R(1,j,d)$, which is equivalent to
the problem of recognizing intersection graphs of $j$-dimensional convex sets
in $\mathbb{R}^d$, for any $j$ and $d$. Furthermore, we point out some trivial
cases of $R(k,j,d)$, and demonstrate that $R(k,j,d)$ is ER-complete for
$j\in\{d-1,d\}$ and $k\geq d$.

Computational Geometry