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 $k\geq d$.

翻译：我们研究了判定给定抽象单纯复形 $K$ 是否为 $\mathbb{R}^d$ 中 $j$ 维凸集神经的 $k$ 骨架的问题。将此问题记为 $R(k,j,d)$。作为主要贡献，我们在这一框架下统一了先前多项工作的结果，并表明这些工作实际上隐含了比明确陈述更强的结论。这使我们能够确定 $R(1,j,d)$ 的复杂性状态（该问题等价于识别 $\mathbb{R}^d$ 中 $j$ 维凸集交图的问题），对于任意 $j$ 和 $d$ 均成立。此外，我们指出了 $R(k,j,d)$ 的某些平凡情形，并证明了当 $j\in\{d-1,d\}$ 且 $k\geq d$ 时，$R(k,j,d)$ 是 ER-完全的。