I have a hyperplane arrangement in $\mathbb{R}^d$ consisting of $d$ families of hyperplanes $F_1,...,F_d$ and a Ball $B$. I want to know in how many pieces the ball $B$ is cut at least. Furthermore I have some extra conditions:
- All hyperplanes intersect $B$.
- For every choice of hyperlanes $H_1\in F_1,...,H_d \in F_d$ the intersection of these hyperplanes is a point in $B$.
- At most $\frac{1}{d}\vert F_i \vert$ planes from the $i$-th family intersect in one point.
My approach until now is to use a dual version of Becks theorem from https://arxiv.org/abs/1607.00048 to get a lower bound of the total number of regions which exist in the hyperplane arrangment. The problem is I am not able to show how many of this regions intersect $B$. Naively I would assume most of them should intersect $B$ since all the planes intersect $B$.
Maybe one could argue with the vertices of the regions, since it is easy to see that each region has at least one vertex. But then I have to find a condition, when a vertex is outside of $B$.
Thanks for your help and ideas, best regards.