Corollary 7.14 . [02XB]

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Corollary 7.14.

Let $\Delta\subset\mathbb{R}^{n}$ be a simplex of dimension $n$ that is the convex hull of points $\nu_{i}$ , $i=0,\dots,n$ , and let $u\in\mathbb{R}^{n}$ such that $\langle u,\nu_{i}\rangle\neq\langle u,\nu_{j}\rangle$ for $i\neq j$ . 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}=n!{\operatorname{vol}}_{n}(\Delta)\sum_{i=0}^{n}\frac{f(\langle u,\nu_{i}\rangle)}{\prod_{j\neq i}\langle\nu_{i}-\nu_{j},u\rangle}.