2 Tidied up the LaTeX

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 /

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

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:

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

In particular, we have:

• 2/3 π2 for n=2
• 8/45 π4 for n=3
• 4/1575 π8 for n=4
1

Building on the nice answer of Guillaume: The integral

\int[-1,1]n \prodi<j |xi2 - xj2| dx1...dxn

has the closed-form evaluation

4n / \prodk≤n \binom{2k}{k}.

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! \prodk≤n πk / ((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