Building on the nice answer of Guillaume: The integral
$$ \int_{[-1,1]^n} \prod\_{i < j} |x_i^2 - x_j^2 | dx_1\dots dx_n $$
has the closed-form evaluation
$$ 4^n \prod_{k \leq n} \binom{2k}{k}^{-1}.$$
This basically follows from the evaluation of the Selberg beta integral Sn(1/2,1,1/2).
Combined with modding out by a typo, we now arrive at the following product formula for the volume of the unit ball of nxn matrices in the matrix norm:
$$ n! \prod_{k\leq n} \frac{ \pi^k }{ ((k/2)! \binom{2k}{k})} .$$
In particular, we have:
- 2/3 π2 for n=2
- 8/45 π4 for n=3
- 4/1575 π8 for n=4
