We shall need the following construction, see Figure 1. Let $\sigma=\sigma_{J}$ be a face of $\Delta$ and $L\subset I$ the set of vertices of $\Delta$ contained in $\sta_{\Delta}(\sigma)$. Consider a rational point $v$ in the relative interior of $\sigma$. Given $0<\varepsilon<1$ rational and $j\in L$ set $e_{j}^{\varepsilon}:=\varepsilon e_{j}+(1-\varepsilon)v$. We shall define a projective simplicial subdivision $\Delta^{\prime}=\Delta^{\prime}(\varepsilon,v)$ of $\Delta$.

To define $\Delta^{\prime}$, first consider a polyhedral subdivision $\Delta^{\varepsilon}=\Delta^{\varepsilon}(v)$ of $\Delta$ leaving the complement of $\sta_{\Delta}(\sigma)$ unchanged. The set of vertices of $\Delta^{\varepsilon}$ is precisely $(e_{i})_{i\in I}\cup(e_{j}^{\varepsilon})_{j\in L}$ and the faces of $\Delta^{\varepsilon}$ contained in $\sta(\sigma)$ are of one of the following types:

• •

  if the convex hull $\Conv(e_{j_{1}},\dots,e_{j_{m}})$ is a face of $\Delta$ containing $\sigma$, then $\Conv(e_{j_{1}}^{\varepsilon},\dots,e_{j_{m}}^{\varepsilon})$ is a face of $\Delta^{\varepsilon}$;

• •

  if $\Conv(e_{j_{1}},\dots,e_{j_{m}})$ is a face of $\Delta$ contained in $\sta(\sigma)$ but not containing $\sigma$, then both $\Conv(e_{j_{1}},\dots,e_{j_{m}})$ and $\Conv(e_{j_{1}},\dots,e_{j_{m}},e_{j_{1}}^{\varepsilon},\dots,e_{j_{m}}^{\varepsilon})$ are faces of $\Delta^{\varepsilon}$.

In a neighborhood of $v$, note that the subdivision $\Delta^{\varepsilon}$ is obtained by scaling $\Delta$ by a factor $\varepsilon$. More precisely, consider the affine map $\psi^{\varepsilon}:\sta(\sigma)\to\sta(\sigma)$ defined by $\psi^{\varepsilon}(w)=\varepsilon w+(1-\varepsilon)v$. Then $\sigma^{\varepsilon}:=\psi^{\varepsilon}(\sigma)$ is the face of $\Delta^{\varepsilon}$ containing $v$ in its relative interior, and $\psi^{\varepsilon}(\sta_{\Delta}(\sigma))=\sta_{\Delta^{\varepsilon}}(\sigma^{\varepsilon})$. In particular, even though $\Delta^{\varepsilon}$ is not simplicial in general, all polytopes of $\Delta^{\varepsilon}$ containing $\sigma^{\varepsilon}$ are simplicial.

We claim that $\Delta^{\varepsilon}$ is projective. To see this, write $v=\sum_{j\in J}s_{j}e_{j}$, with $s_{j}>0$ rational and $\sum s_{j}=1$. For $j\in J$, define a linear function $\ell_{j}$ on $\sum_{i\in I}\mathbf{R}_{+}e_{i}\supset\Delta$ by $\ell_{j}(\sum t_{i}e_{i})=-t_{j}/s_{j}$ and set $h=\max\{\max_{j\in J}\ell_{j},-(1-\varepsilon)\}$. A suitable integer multiple of $h$ is then a strictly convex support function for $\Delta^{\varepsilon}$ in the sense of §3.5.

Now define $\Delta^{\prime}=\Delta^{\prime}(\varepsilon)$ as a simplicial subdivision of $\Delta^{\varepsilon}$ obtained using repeated barycentric subdivision in a way that leaves $\sta_{\Delta^{\varepsilon}}(\sigma^{\varepsilon})$ unchanged. By [KKMS, pp.115–117], $\Delta^{\prime}$ is still projective.

Note that $\sigma^{\prime}:=\sigma^{\varepsilon}$ is the face of $\Delta^{\prime}$ containing $v$ in its relative interior, For $j\in L$ set $e^{\prime}_{j}=e^{\varepsilon}_{j}$. These are the vertices of $\Delta^{\prime}$ contained in $\sta_{\Delta^{\prime}}(\sigma^{\prime})$.