# Equivalent Definitions For Bmo

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 Steinâ€™s Harmonic Analysis.