Inserted: 4 jul 2014
Last Updated: 23 jul 2014
We prove that the Besicovitch Covering Property (BCP) holds for homogeneous distances on the Heisenberg groups whose unit ball centered at the origin coincides with an Euclidean ball. We provide therefore the first examples of homogeneous distances that satisfy BCP in these groups. Indeed, commonly used homogeneous distances, such as (Cygan-)Koranyi and Carnot-Caratheodory distances, are known not to satisfy BCP. We also generalize these previous results showing two criteria that imply the non-validity of BCP, showing that in some sense our examples are sharp. To put another perspective on our result, inspired by an observation of D. Preiss, we prove that in a general metric space with an accumulation point, one can always construct a bi-Lipschitz equivalent distance that does not satisfy BCP.
Keywords: Covering theorems, Heisenberg groups, Homogeneous distances