- journal article metadata
European Mathematical Society Publishing House
2017-04-13 23:45:01
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
19
2017
5
Besicovitch covering property for homogeneous distances on the Heisenberg groups
Enrico
Le Donne
University of Jyväskylä, JYVÄSKYLÄ, FINLAND
Séverine
Rigot
Université de Nice Sophia Antipolis, NICE CEDEX 2, FRANCE
Covering theorems, Heisenberg groups, homogeneous distances
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.
Measure and integration
Abstract harmonic analysis
Calculus of variations and optimal control; optimization
1589
1617
10.4171/JEMS/701
http://www.ems-ph.org/doi/10.4171/JEMS/701