Let $\Delta\subset\mathbb{R}^{n}$ be a polytope of dimension $n$ and $u\in\mathbb{R}^{n}$. Then, for any $f\in\mathscr{C}^{n}(\mathbb{R})$,

	$\displaystyle\int_{\Delta}f^{(n)}(\langle u,x\rangle)\,\text{\rm d}\operatorname{vol}_{n}$	$\displaystyle=\sum_{V\in\Delta(u)}\big(C(\Delta,u,V)(z)\cdot f(z+\langle u,V\rangle)\big)^{(\dim(V))}(0)$
(7.4)			$\displaystyle=\sum_{V\in\Delta(u)}\sum_{k\geq 0}C_{k}(\Delta,u,V)f^{(k)}(\langle u,V\rangle).$

The coefficients $C_{k}(\Delta,u,V)$ are uniquely determined by this identity.