Let $C_k \in \mathbb{R}^n$ for $k \in \{1, ..., m\}$ and $C_0^1, C_0^2 \in \mathbb{R}^n$. Consider the polytopes
$$ \mathcal{P}_1 = \{x | \max_{k} \|x - C_k\|^2 - r_k^2 - \|x - C_0^1\|^2 \leq -R_1^2\}$$
and
$$ \mathcal{P}_2 = \{x | \max_{k} \|x - C_k\|^2 - r_k^2 - \|x - C_0^2\|^2 \leq -R_2^2\}$$
How to compute the complexity of checking if the two polytopes have a common vertex? How about the complexity of finding one, if exists?