For $n\ge 2\in\mathbb N$, let $S\_n$ be the volume of a $(n-1)$ dimensional solid which satisfies 
$$\sum\_{i=1}^{n}x\_i=0, |x\_i|\le1\ (i=1,2,\cdots,n).$$


Then, here is my question.



> 
> **Question** : Can we represent $S\_n$ by $n$ ?
> 
> 
> 


**Remark** : This question has been [asked previously on math.SE](https://math.stackexchange.com/questions/555676/about-a-solid-which-satisfies-sum-i-1nx-i-0-x-i-le1-i-1-2-cdots-n) without receiving any answers.


**Motivation** : I've been interested in this simple question. I've got the followings : 


$$S\_2=2\sqrt 2, S\_3=3\sqrt 3, S\_4=\frac{32}{3}.$$


$S\_4$ is the volume of a regular octahedron, whose edge length is $\sqrt 8$, which passes through the following six points : 


$$(1,1,-1,-1),(1,-1,1,-1),(1,-1,-1,1),(-1,1,1,-1),(-1,1,-1,1),(-1,-1,1,1).$$


However, I don't have any good idea for $n$ in general. Can anyone help?



About a solid which satisfies $\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n)$
This question (in a much more general form) is answered in [this preprint by Marichal and Mosinghoff.](http://arxiv.org/abs/math/0607715) They point out that the answer to your question actually goes back to Polya's PhD thesis.


