I am trying to solve the following problem : Given a set $A$ in $\mathbb{R}^n$ and a point $p$ , I want to find a convex subset of $A$, call it $C$, such that $p$ is in $C$ and random walk starting at $p$ will have the maximum time-to-exit from $C$ (time-to-exit is the expected value of r.v. $X$ = number of time steps till random walk left $C$).
Any sources, ideas, directions will be helpful.
This is not h.w - the solution for that problem is relevant for my research. I didn't find any articles (or other sources) which deal with this problem. I tried to find such $C$ by myself but couldn't get any satisfied results.