Consider the $2^n$ hyperplanes in $\mathbb{R}^{n+1}$ passing through the origin one for each $x ∈ \left \{0, 1\right \}^n$ with normal $(x, 1)$. These hyperplanes divide $\mathbb{R}^{n+1}$ into $2\sum_{k=0}^{n} \binom{2^n-1}{k}$ connected regions.
Say regions "neighbors" if they have a common hyperplane.
How many neighboring regions do the selected region have?
I want to prove that the neighbors $\geq n$.