Suppose I had an open set $X\in \mathbb{R}^n$, and I wish to partition it into two sets $U$ and $V$ where $U$ is the largest possible convex set contained in $X$ and $V$ is some "residual" open set, and $X=U\bigcup V$.
How would I find $U,V$? Also would the answer change is $U$ is closed convex vs open convex, or if the closure $cl(U)\subset X$ vs. $cl(U)$ NOT contained in $X$.
Edit for my purposes, I am mostly interested in bounded sets, but I would also like to study the unbounded case.