# 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?

MathOverflow Asked by Christian Gaetz on February 12, 2021

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.

Answered by Christian Stump on February 12, 2021

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