Such a complex $\Pi$ is determined by a function $v:\Delta_{1}\cap\mathbb{Z}^{n+1}\to\mathbb{R}$, i.e. by $n+2$ numbers $a_{1},\dots,a_{n+1},b\in\mathbb{R}$. Recall (see Example 2) that $\Pi$ is the corner locus $L_{v}(x_{1},\dots,x_{n+1})=\max\{x_{j}-a_{j},-b\}$. If $a_{j}=b=0$ for all $j$ than $\Pi=\Sigma_{1}$. Adding the same real number to all numbers does not change $\Pi$. Changing $a_{j}$ by $t$ results in a translation by $t$ in the direction of $x_{j}$. ∎