Volume of a convex hull in $n$ dimensions

Question

Given $a_1,\cdots,a_n$, s.t. $a_i>0$ for all $i$, consider the set of points: $$ P=\{ \hat{\mathbf{p}}_i= \langle p_1,\cdots,p_n \rangle \;|\; p_i=a_i, p_j = 0 \text{ for } i\neq j \} $$

The set of points $P$ define the hyperplane:

$$ \frac{1}{a_1}x_1+\cdots+\frac{1}{a_n}x_n=1 $$

Let $C$ be the convex hull of the points $P\cup \{\langle 0,\cdots,0 \rangle\}$.

1. Does $C$ have a special name in geometry? That's a special case of a simplex, right?
2. How can I compute the volume of $C$.

Answer 1

1. Yes, this is a simplex.

2. The volume is $1/n!$ times the determinant of the matrix with rows $\hat{\mathbf p}_1, \dotsc, \hat{\mathbf p}_n.$