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 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:

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

In particular, we have:

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

Building on the nice answer of Guillaume: The integral

$$ \int_{[-1,1]^n} \prod_{i < j} \left| x_i^2 - x_j^2 \right| \, dx_1 \dots dx_n $$

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 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:

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

In particular, we have:

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

Building on the nice answer of Guillaume: The integral

\int_[-1,1]ⁿ \prod_i<j |x_i² - x_j²| dx₁...dx_n

has the closed-form evaluation

4ⁿ / \prod_k≤n \binom{2k}{k}.

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 / ((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

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 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:

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

In particular, we have:

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