Let $u$ denote a fixed unit vector in $\mathbb{R}^n$ and $g$ a standard Gaussian vector (in $\mathbb{R}^n$).
Consider the map $$ f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle, $$ defined for real symmetric positive definite $X \succ 0$.
Question: Consider $$ F^\star_n = \sup_{X \succ 0, \mathrm{tr}(X) = 1} f_n(X). $$ Is it possible to determine $F^\star_n$ as a function of the dimension $n$? If not, reasonably sharp bounds (inequalities) would also be interesting.
If $n = 1$, then it is trivial to see that $F^\star_n = \mathbb{E}[(g^2 + 1)^{-1}]$ where $g$ is a standard scalar Normal variate.
More generally, the Sherman-Morrison formula gives us $$ f_n(X) = \langle Xu, u \rangle - \mathbb{E} \Big[\frac{\langle Xu, g\rangle^2}{1 + \langle X g, g \rangle}\Big] $$ However, maximizing this over unit trace $X \succ 0$ is not so clear to me.
