Consider an $n$-dimensional random vector $X$ whose probability distribution $P_{\mu,\Sigma}$ is a multivariate normal law of mean vector $\mu\in\mathbb{R}^n$ and covariance matrix $\Sigma\in\mathbb{R}^{n\times n}$.
Let $V$ be a $n$-simplex of $\mathbb{R}^n$.
Problem: Find $ \mu^* = {argmax}_{\mu\in\mathbb{R}^n} \left( P_{\mu,\Sigma}(X\in V) \right) $
I am not sure that the solution to this problem has a closed-form.
Intuitively I would say that if the covariance matrix is the identity the solution is the center of the simplex but I am not able to prove that.
Taking the gradient of the probability to maximize and using the divergence theorem yields a sum of $n+1$ vectors. These vectors are normal to the faces of $V$ and are weighted by the probabilities of normal laws on these faces. Therefore those weights don't have a closed-form, making hard the search for a solution by taking that gradient equal to $0$.