Journal of the European Mathematical Society
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Published online: 2017-04-10
Besicovitch covering property for homogeneous distances on the Heisenberg groupsEnrico Le Donne and Séverine Rigot (1) University of Jyväskylä, Finland
(2) University of Nice Sophia Antipolis, France
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 a Euclidean ball. We thus provide the first examples of homogeneous distances that satisfy BCP on these groups. Indeed, commonly used homogeneous distances, such as (Cygan–)Korányi and Carnot–Carathéodory distances, are known not to satisfy BCP. We also generalize those previous results by giving two geometric criteria that imply the non-validity of BCP and showing that in some sense our examples are sharp. To put our result in another perspective, inspired by an observation of D. Preiss, we prove that in a general metric space with an accumulation point, one can always construct bi-Lipschitz equivalent distances that do not satisfy BCP.
Keywords: Covering theorems, Heisenberg groups, homogeneous distances
Le Donne Enrico, Rigot Séverine: Besicovitch covering property for homogeneous distances on the Heisenberg groups. J. Eur. Math. Soc. 19 (2017), 1589-1617. doi: 10.4171/JEMS/701