From the finiteness condition in Definition 1 we have that the complex $\Pi^{\prime}=(\Pi-R\hskip-5.0pt\stackrel{{\scriptstyle\to}}{{v}})\cap L\subset L$ does not depend on the choice of $R>0$ and $\stackrel{{\scriptstyle\to}}{{v}}$ as long as $\stackrel{{\scriptstyle\to}}{{v}}$ is supporting and $R$ is sufficiently large. The proof of Proposition 1.4 ensures that $\Pi^{\prime}$ is a dual $\Delta^{\prime}$-complex. If $\Pi$ is maximal then it is dual to a triangulation of $\Delta$ into simplices of minimal volume. Such a triangulation induces a triangulation into simplices of minimal volume on the faces $\Delta^{\prime}$ and thus $\Pi^{\prime}$ is also maximal. ∎