2 clarification

Yes, O(n) is the n(n-1)/2 dimensional space of orthogonal n by n matrices. Vol(O(n)) is its volume.

The integrand in the answer is simply the Jacobian of the singular value decomposition, {s_ i} is just the ordered set of the singular value and the integration is performed on the subset bounded by 1.

I may just have missed a factor of 1/2^n because of the sign ambiguity in the svd singular values

1

Yes, O(n) is the n(n-1)/2 dimensional space of orthogonal n by n matrices.

The integrand in the answer is simply the Jacobian of the singular value decomposition, {s_ i} is just the ordered set of the singular value and the integration is performed on the subset bounded by 1.