TITLE: Volume of Region in 5D Space
QUESTION [46 upvotes]: I need to find the volume of the region defined by
$$\begin{align*}
a^2+b^2+c^2+d^2&\leq1,\\ 
a^2+b^2+c^2+e^2&\leq1,\\ 
a^2+b^2+d^2+e^2&\leq1,\\ 
a^2+c^2+d^2+e^2&\leq1 &\text{ and }\\
b^2+c^2+d^2+e^2&\leq1.
\end{align*}$$
I don't necessarily need a full solution but any starting points would be very useful.

REPLY [5 votes]: It is clear that the valid region lies between the hyperspheres of radius $r_1=1$ and $r_2=\sqrt{5/4}$. This already gives some trivial bounds, but the upper bound is not tight. 
Lets try to compute the excess volume, beyond the unitary sphere. 
Let fix some $1<r<r_2$. The valid region over that surface corresponds to the points with $|x_i|>\sqrt{r^2-1}$ ($\forall i$). We wish to compute $p(r)$, the proportion of the surface that lies in that region, i.e. the probability that if we throw a random point uniformily over a hypershpere surface, all its coordinates are greater than some value.  This happens to be closely related to other recent question. The approximation I proposed there, using independent  gaussians, can also be applied here, and we can expect to perform better (because we are interested in a more "probable" region). Applying the same reasoning, ($n$ gaussians with variance $r/n$ here, and $\epsilon = \sqrt{r^2-1}$), we get the approximation:
$$p(r) \approx \left[ 1 - erf\left(\sqrt{ n \, \frac{r^2-1}{2 r}}\right)\right]^n$$
and from this we can compute the total volume integrating:
$$ V(n) = V_1(n) + V_e(n) = \frac{\pi^{n/2}}{\Gamma{(\frac{n}{2}+1})} + n \frac{\pi^{n/2}}{\Gamma{(\frac{n}{2}+1})} \int_1^{r_2} p(r) \; r^{n-1} dr$$
where $r_2 = \sqrt{\frac{n}{n-1}}$. Pluggin the above approximation, I get these values:
 n       A       J        Ae      Je
---------------------------------------
 2    3.7664   4.0000   0.6248   0.8584
 3    4.6359   4.6863   0.4471   0.4975
 4    5.2619   5.2746   0.3271   0.3398
 5    5.5004   5.5036   0.2366   0.2398
 6    5.3353   5.3361   0.1676   0.1684
 7    4.8406   4.8408   0.1158   0.1160
 8    4.1366   4.1367   0.0779   0.0780

where column A is my approximation, J are the values from Joriki's answer (Montecarlo integration for  $n>5$). The columns Ae,Je show the respective excess volumes over the unitary hypersphere, they are actually the relevant quantities to be compare to judge the goodness of the approximation. (I must say I'm surprised that gives so good results for small $n$, and the speed of convergence)
Here goes the Octave/Matlab code:
clear
global N=5
r2=sqrt(N/(N-1));

function pp=pp(r)
 global N;
 pp = (1-erf( sqrt( N* (r.^2 - 1)./(2*r) ) )).^N .* r.^(N-1);
endfunction

(pi^(N/2)/gamma(N/2+1)) * ( 1 + quad ("pp",1,r2) * N)