Let $X$ be a random vector with its distribution being uniform on the $\ell_2$ unit ball. Is it true to say that $X\space \space | \space \space||X||_2$ is uniformly distributed on the surface of the sphere of radius $||X||_2$?
Intuitively this result would make sense, but how could I prove this mathematically (if it's true)? If I'm honest, I don't know where to start because I don't understand what the densities of these random variables are. Please help :(