Uniform distribution over an hyper-ellipsoid

Question

Let $\mathbf{X} \in \bf{R}^p$ be a random vector whose elements are uniformly distributed over the hyper-ellipsoid $x^TAx<1$, (where $A$ is a positive-definite matrix). Is it possible to compute the covariance matrix of $\mathbf{X}$, and would it have any relation to $A$?

Answer 1

The error covariance matrix is \begin{equation} \frac{1}{p+2} A^{-1}. \end{equation}