I'am considering an arrangement $\mathcal{H}$ of $n$ hyperplanes in $\mathbb{R}^d$ and a ball $B \subset \mathbb{R}^d$ such that every hyperplane intersects the ball non-trivially. I want to know into how many pieces the ball at least gets cut by the hyperplanes.
There is an obvious lower bound given by $n+1$ if all the planes are parallel. But in this case also the number of regions formed by the arrangement in $\mathbb{R}^d$ is $n+1$ (in other words the number of connected components of $\mathbb{R}^d \setminus \bigcup_{H \in \mathcal{H}} H$).
So is there a possibility to get a lower bound if one knows into how many connected components the total space is cut? Is there any reference to such a question or a similar one?
It is not possible to get a lower bound better than $n+1$. Suppose we have some hyperplane arrangement $\mathcal{H}$ with $k$ regions. By applying an affine transformation to this arrangement, we can make the angles between these hyperplanes as small as we like (say, by stretching along an axis which is not normal to any of the hyperplanes in $\mathcal{H}$), without affecting $k$.
Then, we can just place a ball somewhere along these now-very-well-aligned hyperplanes, such that they do not intersect each other anywhere in the ball.
(Equivalently, and perhaps easier to visualize, if you can find any ellipsoid which intersects each hyperplane without containing any shared intersections, you can make affine transformations to the ambient space to convert that ellipsoid into a sphere while preserving the incidence structure. So you just need to send a very pointy "spear" of an ellipsoid through the hyperplane arrangement while avoiding any lower-dimensional intersections.)