Consider the function

This is a convex piecewise-linear function $\mathbb{R}^{n+1}\to\mathbb{R}$. We define the primitive complex $\Sigma_{n}\subset\mathbb{R}^{n+1}$ as the corner locus of $H$, i.e. the set of points where $H$ is not smooth is $\Sigma_{n}$.

Note that $\Sigma_{n}$ is a balanced proper polyhedral complex in $\mathbb{R}^{n+1}$. Its $k$-cells are formed by the points where at least $n+2-k$ of the functions $0,x_{1},\dots,x_{n+1}$ achieve the value of $H$. In fact, it is easy to see that topologically $\Sigma_{n}$ is the cone over the $(n-1)$-skeleton of the $(n+1)$-simplex. The fact that $\Sigma$ is balanced follows from Proposition 1.2.