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We develop some techniques for studying various versions of the functionspace BMO. Special cases of one of our results give alternative proofs of thecelebrated John- Nirenberg inequality and of related inequalities due to Johnand to Wik. Our approach enables us to pose a simply formulated "geometric"question, for which an affirmative answer would lead to a version of theJohn-Nirenberg inequality with dimension free constants. A more detailedsummary of the main ideas and results of this paper can be found athttp://www.math.technion.ac.il/~mcwikel/bmo/CwikSaghShvaSummary.pdf

A new look at the John-Nirenberg and John-Stromberg theorems for BMO. Lecture Notes