Volume of a convex hull in $n$ dimensions

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 $C$.
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1 Answer

1. Yes, this is a simplex.

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

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Could you elaborate more on #2. How is it derived? –  Helium Dec 9 '13 at 19:02
The linear transformation which maps $0$ to $0$ and the standard basis to $p_1, \dotsc, p_n$ multpilies volume by its determinant, and the volume of the standard simplex (one with vertices at $0, e_1, \dotsc, e_n$ has volume $1/n!,$ which you can prove by induction by integrating... –  Igor Rivin Dec 9 '13 at 19:06
Nice! I got it. Could you please include your comment in the answer and add a couple of references, if possible. I'll tag it as accepted then. I also made a small edit in your answer. Thanks. –  Helium Dec 9 '13 at 19:24