Let $\Delta\subset\mathbb{R}^{n}$ be a polytope of dimension $n$ and $u\in\mathbb{R}^{n}$. Let $V\in\Delta(u)$ and $k\geq 0$.

1. (1)

   The coefficient $C_{k}(\Delta,u,V)$ is homogeneous of weight $k-n$, in the sense that, for $\lambda\in\mathbb{R}^{\times}$,

   $$C_{k}(\Delta,\lambda u,V)=\lambda^{k-n}C_{k}(\Delta,u,V).$$

2. (2)

   The coefficients $C_{k}(\Delta,u,V)$ satisfy the vector relation

   (7.9)

   $$C_{k}(\Delta,u,V)\cdot u=-\sum_{F}C_{k}(F,\pi_{F}(u),V\cap F)\cdot u_{F},$$

   where the sum is over the facets $F$ of $\Delta$.

3. (3)

   Let $\Delta_{1},\Delta_{2}\subset\mathbb{R}^{n}$ be two polytopes of dimension $n$ intersecting along a common facet and such that $\Delta=\Delta_{1}\cup\Delta_{2}$. Then $V\cap\Delta_{i}=\emptyset$ or $V\cap\Delta_{i}\in\Delta_{i}(u)$ and

   $$C_{k}(\Delta,u,V)=C_{k}(\Delta_{1},u,V\cap\Delta_{1})+C_{k}(\Delta_{2},u,V\cap\Delta_{2}).$$