As set systems, hypergraphs are omnipresent and have various representations. In a geometric representation of a hypergraph $H=(V,E)$, each vertex $v\in V$ is a associated with a point $p_v\in \mathbb{R}^d$ and each hyperedge $e\in E$ is associated with a connected set $s_e\subset \mathbb{R}^d$ such that $\{p_v\mid v\in V\}\cap s_e=\{p_v\mid v\in e\}$ for all $e\in E$. We say that a given hypergraph $H$ is representable by some (infinite) family $\mathcal{F}$ of sets in $\mathbb{R}^d$, if there exist $P\subset \mathbb{R}^d$ and $S \subseteq \mathcal{F}$ such that $(P,S)$ is a geometric representation of $H$. For a family $\mathcal{F}$, we define RECOGNITION($\mathcal{F}$) as the problem to determine if a given hypergraph is representable by $\mathcal{F}$. It is known that the RECOGNITION problem is ER-hard for halfspaces in $\mathbb{R}^d$. We study the families of balls and ellipsoids in $\mathbb{R}^d$, as well as other convex sets, and show that their RECOGNITION problems are also ER-complete. This means that these recognition problems are equivalent to deciding whether a multivariate system of polynomial equations with integer coefficients has a real solution.

翻译：作为集合系统，超图无处不在且具有多种表示形式。在超图 $H=(V,E)$ 的几何表示中，每个顶点 $v\in V$ 对应一个点 $p_v\in \mathbb{R}^d$，每条超边 $e\in E$ 对应一个连通集 $s_e\subset \mathbb{R}^d$，使得对所有 $e\in E$ 有 $\{p_v\mid v\in V\}\cap s_e=\{p_v\mid v\in e\}$。对于给定的超图 $H$，若存在 $\mathbb{R}^d$ 中的点集 $P\subset \mathbb{R}^d$ 与某个（无限）集合族 $\mathcal{F}$ 的子集 $S \subseteq \mathcal{F}$，使得 $(P,S)$ 构成 $H$ 的几何表示，则称 $H$ 可由 $\mathcal{F}$ 表示。对集合族 $\mathcal{F}$，我们定义识别问题RECOGNITION($\mathcal{F}$)：判定给定超图是否可由 $\mathcal{F}$ 表示。已知在 $\mathbb{R}^d$ 中，半空间族的RECOGNITION问题是ER-困难的。本文研究 $\mathbb{R}^d$ 中球族、椭球族及其他凸集的RECOGNITION问题，证明它们均为ER-完全的。这意味着这些识别问题等价于判定一个带整数系数的多元多项式方程组是否存在实数解。