Let $\Delta\subset\mathbb{R}^{n}$ be a simplex of dimension $n$ and let $\ell\colon\mathbb{R}^{n}\to\mathbb{R}$ be an affine function which is non-negative on $\Delta$. Write $\ell(x)=\langle u,x\rangle-\lambda$ for some vector $u$ and constant $\lambda$. Then $\displaystyle\frac{1}{\operatorname{vol}_{n}(\Delta)}\int_{\Delta}\ell(x)\log(\ell(x))\,\text{\rm d}\operatorname{vol}_{n}$ equals

(7.16)

$$\sum_{V\in\Delta(u)}\sum_{\beta^{\prime}}\binom{n}{n-|\beta^{\prime}|}\frac{\ell(V)\left(\log(\ell(V))-\sum_{j=2}^{|\beta^{\prime}|+1}\frac{1}{j}\right)}{(|\beta^{\prime}|+1)\prod_{\nu\notin V}\left(-\big(\frac{\ell(\nu)}{\ell(V)}-1\big)^{\beta^{\prime}_{\nu}}\right)},$$

where the second sum is over $\beta^{\prime}\in(\mathbb{N}^{\times})^{n-\dim(V)}$ with $|\beta^{\prime}|\leq n$ and the product is over the $n-\dim(V)$ vertices $\nu$ of $\Delta$ not in $V$. In case $\ell(x)$ is the defining equation of a hyperplane containing a facet $F$ of $\Delta$,

(7.17)

$$\frac{1}{\operatorname{vol}_{n}(\Delta)}\int_{\Delta}\ell(x)\log(\ell(x))\,\text{\rm d}x=\frac{\ell(\nu_{F})}{n+1}\bigg(\log(\ell(\nu_{F}))-\sum_{j=2}^{n+1}\frac{1}{j}\bigg),$$

where $\nu_{F}$ denotes the unique vertex of $\Delta$ not contained in $F$.