Connected components of a given set

Question

Let $H_1,H_2,H_3, H_4$ be four hyperplanes in $\mathbb{R}^3$ . I need to find the maximum number of connected components of $\mathbb{R}^3-H_1\cup...\cup H_4$. I have no idea how to visualise it. This is easy to do in $\mathbb{R}^2$ . Also can we construct a general formula for maximum number of connected componets In $\mathbb{R}^n$ with $m$ hyperplanes.

Answer 1

Yes, this is known as the cake number.

In general, the maximum number of components into which $m$ hyperplanes inside $\mathbb R^n$ cut $\mathbb R^n$ is:

$\binom{m}{n}+\binom{m}{n-1}+\dots+ \binom{m}{1}+\binom{m}{0}$.