Question

Building on the nice answer of Guillaume: The integral

$$ \int_{[-1,1]^{nint_{[-1,1]^n} \prodi<jprod_{i < j} |xi2x_i^2 - xj2x_j^2 | dx1...dxndx_1\dots dx_n $$}}

has the closed-form evaluation

4ⁿ /

$$ 4^n \prod_{k≤nprod_{k \binom{2k}{k}.leq n} \binom{2k}{k}^{-1}.$$}

This basically follows from the evaluation of the Selberg beta integral S_n(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:

$$ n! \prod_k≤n π^k / prod_{k\leq n} \frac{ \pi^k }{ ((k/2)! \binom{2k}{k}).binom{2k}{k})} .$$

In particular, we have:

• 2/3 π² for n=2
• 8/45 π⁴ for n=3
• 4/1575 π⁸ for n=4