Besicovitch Covering Property for homogeneous distances in the Heisenberg groups

Our main result is a positive answer to the question whether one can find homogeneous distances on the Heisenberg groups that have the Besicovitch Covering Property (BCP). This property is well known to be one of the fundamental tools of measure theory, with strong connections with the theory of differentiation of measures. We prove that BCP is satisfied by the homogeneous distances whose unit ball centered at the origin coincides with an Euclidean ball. Such homogeneous distances do exist on any Carnot group by a result of Hebisch and Sikora. In the Heisenberg groups, they are related to the Cygan-Koranyi (also called Koranyi) distance. They were considered in particular by Lee and Naor to provide a counterexample to the Goemans-Linial conjecture in theoretical computer science. To put our result in perspective, we also prove two geometric criteria that imply the non-validity of BCP, showing that in some sense our example is sharp. Our first criterion applies in particular to commonly used homogeneous distances on the Heisenberg groups, such as the Cygan-Koranyi and Carnot-Caratheodory distances that are already known not to satisfy BCP. To put a different perspective on these results and for sake of completeness, we also give a proof of the fact, noticed by D. Preiss, that in a general metric space, one can always construct a bi-Lipschitz equivalent distance that does not satisfy BCP.