\cdots & 0 \\
        1 & 1 & 1 & \cdots & 0 \\
        \vdots & \vdots & \vdots & \ddots & \vdots \\
        1 & 1 & 1 & \cdots & 1 \\
    \end{bmatrix}$$ to each vertex of $\nabla_{C_N}$. One reason that this is desirable is that it becomes clear $P_N$ is totally unimodular, that is, the matrix formed by placing the vertices of $P_N$ as the columns is a totally unimodular matrix. Since $\mathbf{0}$ is the average of all vertices of $P_N$, it is an interior point of $P_N$. Thus, a unimodular triangulation of the boundary of $P_N$ will induce a unimodular triangulation of $P_N$ itself, where the simplices are of the form $\operatorname{conv}\{\mathbf{0} \cup \Delta \}$ where $\Delta$ is a simplex in the triangulation of the boundary. This will be our strategy, since the number of simplices in a unimodular triangulation of a polytope is identical to the normalized volume of the polytope.

When $N$ is odd, then $P_N$ is called a *del Pezzo* polytope. In this case, it is known [@Nill] that $P_N$ is simplicial, that is, every facet is a simplex. Together with $P_N$ being totally unimodular, its normalized volume is therefore equal to the 