Suppose I have a (finite, real, central, essential) hyperplane arrangement $\mathcal{H}$ such that all regions "have the same shape": for any two regions $R,R'$, there is an orthogonal transformation taking $R$ to $R'$ (these transformations are not required to do anything nice to the rest of the arrangement). Is $\mathcal{H}$ necessarily a reflection arrangement?



Hyperplane arrangements whose regions all have the same shape
This is a known open problem (for isometric regions), which, as far as I know, is still not settled.


The dimension 3 case was proved affirmatively in <https://arxiv.org/abs/1501.05991>, where also some history of the question is outlined. I am not aware of any progress since.


