In case $\Delta$ is a simplex, its aggregates in a given direction $u\in\mathbb{R}^{n}$ are some of its faces and the corresponding coefficients can be made explicit. Indeed, they satisfy the linear system

$$\sum_{V\in\Delta(u)}\sum_{k=0}^{\min\{i,\dim(V)\}}C_{k}(\Delta,u,V)\frac{\langle u,V\rangle^{i-k}}{(i-k)!}=\begin{cases}0&\mbox{for }i=0,\dots,n-1,\\ {\rm Vol}_{n}(\Delta)&\mbox{for }i=n.\end{cases}$$

This system has as many unknowns as equations and might be solved using Cramer’s rule. These coefficients admit the closed formula below, which the reader might check using the recurrence relation (7.9):

(7.11)

$$C_{k}(\Delta,u,V)=(-1)^{\dim(V)-k}\frac{n!}{k!}{\operatorname{vol}}_{n}(\Delta)\sum_{|\beta|=\dim(V)-k}\prod_{\nu\notin V}\langle V-\nu,u\rangle^{-\beta_{\nu}-1},$$

where the products are over the vertices $\nu$ of $\Delta$ not lying in $V$ and the sum is over the tuples $\beta$ of non negative integers of length $\dim(V)-k$, indexed by those same vertices of $\Delta$ that are not in $V$, that is, $\beta\in\mathbb{N}^{n-\dim(V)}$ and $|\beta|=\dim(V)-k$. In case $V=\nu_{0}$ is a vertex of $\Delta$, the above formula reduces to

(7.12)

$$C_{0}(\Delta,u,\nu_{0})=n!{\operatorname{vol}}_{n}(\Delta)\prod_{\nu\neq\nu_{0}}\langle\nu_{0}-\nu,u\rangle^{-1}.$$

Suppose that the simplex is presented as the intersection of $n+1$ halfspaces as $\Delta=\bigcap_{i=0}^{n}\{x\in\mathbb{R}^{n}|\,\langle u_{i},x\rangle-\lambda_{i}\geq 0\}$ for some $u_{i}\in\mathbb{R}^{n}$ and $\lambda_{i}\in\mathbb{R}$. Up to a reordering, we can assume that $u_{0}$ is normal to the unique face of $\Delta$ not containing $\nu_{0}$ and that $\det(u_{1},\dots,u_{n})>0$. Then the above coefficient can be alternatively written as

(7.13)

$$C_{0}(\Delta,u,\nu_{0})=\frac{\det(u_{1},\dots,u_{n})^{n-1}}{\prod_{i=1}^{n}\det(u_{1},\dots,u_{i-1},u,u_{i+1},\dots,u_{n})}.$$