As set systems, hypergraphs are omnipresent and have various representations ranging from Euler and Venn diagrams to contact representations. In a geometric representation of a hypergraph $H=(V,E)$, each vertex $v\in V$ is 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 $F$ of sets in $\mathbb{R}^d$, if there exist $P\subset \mathbb{R}^d$ and $S \subseteq F$ such that $(P,S)$ is a geometric representation of $H$. For a family F, we define RECOGNITION(F) as the problem to determine if a given hypergraph is representable by F. It is known that the RECOGNITION problem is $\exists\mathbb{R}$-hard for halfspaces in $\mathbb{R}^d$. We study the families of translates of balls and ellipsoids in $\mathbb{R}^d$, as well as of other convex sets, and show that their RECOGNITION problems are also $\exists\mathbb{R}$-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，从欧拉图与文氏图到接触表示。在超图 $H=(V,E)$ 的几何表示中，每个顶点 $v\in V$ 关联于 $\mathbb{R}^d$ 中的一个点 $p_v$，每个超边 $e\in E$ 关联于一个连通集 $s_e\subset \mathbb{R}^d$，使得对所有 $e\in E$ 有 $\{p_v\mid v\in V\}\cap s_e=\{p_v\mid v\in e\}$。我们称给定超图 $H$ 能被 $\mathbb{R}^d$ 中某（无限）集合族 $F$ 表示，若存在 $P\subset \mathbb{R}^d$ 和 $S \subseteq F$，使得 $(P,S)$ 是 $H$ 的一个几何表示。对于集合族 $F$，我们将 RECOGNITION(F) 定义为判定给定超图是否可被 $F$ 表示的问题。已知对于 $\mathbb{R}^d$ 中的半空间，RECOGNITION 问题是 $\exists\mathbb{R}$-困难的。我们研究 $\mathbb{R}^d$ 中球与椭球的平移族以及其他凸集的情形，并证明其 RECOGNITION 问题同样是 $\exists\mathbb{R}$-完全的。这意味着这些识别问题等价于判定一个具有整数系数的多元多项式方程组是否存在实数解。