  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.
