There are many questions conserning an upper bound on the number of regions in an arrangement of hyperplanes in general position, but I'm interested in a lower bound for hyperplanes with some extra conditions.
Let $A_1,...,A_k$ be sets of each $s$ parallel hyperplanes in $\mathbb{R}^n$ and let $d(n,k,s)$ denote the minimum number of regions in such an arangement. We assume that hyperplanes in different $A_i$ are not parallel. Do we know a lower bound to this?
I think $d(n,k,s)=d(2,k,s)$ holds. Further it is clear that $d(n,1,s)=s+1$.
Any idea or source for some more information would helps. Best regards and many thanks in advance.
Here are my own thoughts in dimension two: The parallels from $A_1$ divide the plane in $s+1$ regions. Adding a line from $A_2$ splits this regions, so after adding all the lines form $A_2$ we get $(s+1)+ s \cdot (s+1)$ regions. This is best we can get also for the over $A_i$, so in conclusion we get $(s+1)+ (k-1) \cdot s \cdot (s+1)$ regions. Question: Is this optimal? What if we in dimension 3 we assume that not all planes contain one parallel direction?