Let $f_1, \dots, f_j$ be convex functions from $\mathbb{R}^n \to \mathbb{R}$. I am trying to develop a test that decides whether or not the set $\{x | f_1(x) \le k_1\} \cup \dots \cup \{x | f_n(x) \le k_n\}$ has a hole in it of any size (the alternative is that the set is homeomorphic to the unit ball, maybe plus a few lower-dimensional "fingers").

Is anything known about this problem?

Editing in some extra information that is specific to my particular use for this algorithm. In my algorithm, $n$ of the convex sets that I'm unioning together are the coordinate planes (i.e. $\{x | x_j = 0\}$), and there are exactly $n$ additional convex sets that I care about (so $j = 2n$).