An arrangement of polytopes (analogous to an arrangement of hyperplanes) $\mathcal{P}$ is just a set of polytopes in $\mathbb{R}^d$. Suppose $|\mathcal{P}| = n$.
Then a cell is a connected component of $\mathbb{R^d} \setminus \bigcup_{p \in \mathcal{P}} p$. What is the maximum number of cells in an arrangement of polytopes?
I know that the maximum number of cells in an arrangement of $n$ hyperplanes in $\mathbb{R}^d$ is $\phi_d(n) := \sum_{k=0}^d \binom{n}{d}$ if $d < n$ and $2^n$ o.w. . Is there something similar for arrangements of polytopes?