The space of functions of bounded mean oscillation $BMO$ is defined by the BMO norm

But an equivalent definition is to take the sup over balls instead of cubes. Previously I wondered what other shapes gave an equivalent norm.

Proposition: Suppose $D \subset \mathbb{R}^n$ is a open set such that there exists $0 < r_1 < r_2 < \infty$ such that

then the norm given by

is equivalent to the BMO norm. The set $A_D$ is $D$ under any uniform scaling, rotations transtions or composition thereof.

I have not yet proven this, but I think it should be possible by adapting ideas from Stein’s Harmonic Analysis.