@@ -694,8 +694,9 @@ two functions $\iota,\omega:V(G)\to 2^{\mathbb{R}^d}$ such that for some orderin
\item if $i<j$ and $v_iv_j\in E(G)$, then $\omega(v_j)\cap\iota(v_i)\neq\emptyset$.
\end{itemize}
We say that the representation has \emph{thickness at most $t$} if for every point $x\in\mathbb{R}^d$, there
exist at most $t$ vertices $v\in V(G)$ such that $x\in\iota(v)$.
exist at most $t$ vertices $v\in V(G)$ such that $x\in\iota(v)$. For example, if $f$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^d$, then $(f,f)$ is a $1$-comparable envelope representation of $G$ in $\mathbb{R}^d$ of thickness at most $2^d$.
%If $f$ is a touching representation of $G$ by balls in $\mathbb{R}^d$ and $\omega(v)$ is the smallest axis-aligned hypercube containing $f(v)$, then there exists a positive integer $s_d$ depending only on $d$ such that $(f,\omega)$ is an $s_d$-comparable envelope representation of $G$ in $\mathbb{R}^d$ of thickness at most $2$.
