Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $C$.
share|cite|improve this question
up vote 2 down vote accepted
  1. Yes, this is a simplex.

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.