# Hyperplane arrangements whose regions all have the same shape

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?

• This is more restricted than what you ask, but in this preprint I prove that if the arrangement is transitive on the regions, then it is a reflection arrangement (though it is proven via a dual formulation with zonotopes, see also this question). You probably know that an arrangement with congruent (aka. same-shaped) regions must be simplicial. As far as I know, no classification of simplicial arrangements exists beyond 3D (and I am not sure whether 3D is settled). – M. Winter Nov 12 '20 at 15:32