ScalingStacks

Notation. Denote the vertices of $\partial\Delta_{\lambda}^{\vee}$ as $w_{0},\ldots,w_{n+1}$, which coincide with the outward normal vectors because $\partial\Delta_{\lambda}^{\vee}=-\partial\Delta^{\vee}$. Denote the vertices of $\Delta$ as $m^{0},\ldots,m^{n+1}$, so that

$$\langle w_{i},m^{j}\rangle=\begin{cases}1,\quad&i\neq j,\\ -(n+1),\quad&i=j.\end{cases}$$

Let $\text{Star}(w_{i})$ be the star of $w_{i}$ in the barycentric subdivision of $\partial\Delta_{\lambda}^{\vee}$. Let $Sing\subset\widetilde{Sing}$ be the subset of points not contained in the interior of any of these stars. The affine structure on $\partial\Delta_{\lambda}^{\vee}\setminus\widetilde{Sing}$ extends to $\partial\Delta_{\lambda}^{\vee}\setminus Sing$, by decreeing that on the interior of $\text{Star}(w_{i})$ we use the coordinates for the chart $U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$. As $Sing$ has codimension two inside $\partial\Delta_{\lambda}^{\vee}$, this makes $\partial\Delta_{\lambda}^{\vee}$ into a singular affine manifold.