Let $B=\tau_{1}\cup\tau_{2}$ be the union of two $n$-dimensional simplices along a face of codimension one. Assume we are given a $\mathbb{Z}$-affine structure on $B$, compatible with those on the $\tau_{i}$’s.

Suppose there exists a sequence of (weighted) star subdivisions of $\tau_{1}$ such that $B^{\prime}\coloneqq\Star(\tau_{1}\cap\tau_{2})$ (with respect to this subdivision) can be embedded in $\mathbb{R}^{n}$ compatibly with the $\mathbb{Z}$-affine structure. Then this embedding extends to $B$, and the $\mathbb{Z}$-affine structure on $B$ is uniquely recovered by this embedding.