A geometrical/combinatorical question with implications for the John–Nirenberg inequality for BMO functions
The first and last sections of this paper are intended for a general mathematical audience. In addition to some very brief remarks of a somewhat historical nature, we pose a rather simply formulated question in the realm of (discrete) geometry. This question has arisen in connection with a recently developed approach for studying various versions of the function space $BMO$. We describe that approach and the results that it gives. Special cases of one of our results give alternative proofs of the celebrated John–Nirenberg inequality and of related inequalities due to John and to Wik. One of our main results is that an affirmative answer to the above question would lead to a version of the John–Nirenberg inequality with “dimension free” constants.