Let $q_1,\dots,q_n$ be positive integer numbers and let $$ \mathcal P=\{(x_1,\dots,x_n)\in\mathbb R^n: |x_1|+\dots+|x_n|\leq 1,\; x_1q_1+\dots x_nq_n=0)\}. $$
What is the volume of $\mathcal P$? I need an idea to compute it.
Of course, I mean the $(n-1)$-volume since $\mathcal P$ has dimension $n-1$.