Lemma 1.2 . [014Z]

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Lemma 1.2.

Given $b_{0},\dots,b_{p}\in{\mathbb{N}}^{*}$ , view

\sigma=\left\{w\in{\mathbb{R}}_{+}^{p+1}\mid\sum_{i=0}^{p}b_{i}w_{i}=1\right\}

as a ${\mathbb{Z}}$ -simplex. Then $b_{\sigma}=\gcd(b_{i})$ , and

\operatorname{Vol}(\sigma)=\frac{b_{\sigma}}{p!\prod_{i}b_{i}}.