The complex $\Pi^{\prime}$ from Proposition 1.12 can be obtained in the following way. Let $L\subset\mathbb{R}^{n+1}$ be the linear $(k+1)$-subspace parallel to the face $\Delta^{\prime}$. Let $\stackrel{{\scriptstyle\to}}{{v}}$ be a supporting vector at $\Delta^{\prime}$. For a sufficiently large $R>0$ we have $\Pi^{\prime}=(\Pi-R\hskip-5.0pt\stackrel{{\scriptstyle\to}}{{v}})\cap L$.