I am generating random vectors $X_1, \dots, X_N$ from a $d$-dimensional multivariate normal $\text N(\mu, \Sigma)$. I would like to know what is the probability that a given point $y \in R^d$ falls within the convex hull of the sample (N > d).

I can't find any result concerning this problem, apart from this answer which covers only a specific point in $R^d$ (the mean). Is anybody aware of any work on this topic?

My final aim is finding the point $y$ at which the $P(y \in \text{ConHull}(X_1, \dots, X_N))$ is maximal. Thanks for any suggestion.