I want to choose a random vector in high dimensions such that it all directions have the same uniform chance (i.e. isotropic in all directions). My current idea is the following algorithm:

1. sample v from a high dim multivariate Gaussian
2. then normalize that vector by dividing by its norm

I believe this is sufficient but don't know how to prove it. I know that high dimension stats can have weird behaviours (like most of the volume of a Gaussian is at the "edge") so I wanted to make sure I didn't miss anything trivial but important. Anyone know if this is correct? If it is how do I prove it?

What I have in mind in pytorch looks as follows:

v = MultivariateNormal(torch.zeros(2), torch.eye(2))
v = v/v.norm(2)

I want to choose a random vector in high dimensions such that it all directions have the same uniform chance (i.e. isotropic in all directions). My current idea is the following algorithm:

1. sample v from a high dim multivariate Gaussian
2. then normalize that vector by dividing by its norm

I believe this is sufficient but don't know how to prove it. I know that high dimension stats can have weird behaviours (like most of the volume of a Gaussian is at the "edge") so I wanted to make sure I didn't miss anything trivial but important. Anyone know if this is correct? If it is how do I prove it?

What I have in mind in pytorch looks as follows:

v = MultivariateNormal(torch.zeros(10000), torch.eye(10000))
v = v/v.norm(10000)