Remarks about Besicovitch covering property in Carnot groups of step 3 and higher

2015/03/31 by Enrico Le Donne, Donne, Enrico Le, Severine Rigot +1
Mathematics · #28C15 #43A80 #49Q15 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:28C15 #msc:43A80 #msc:49Q15

paper · pdf · doi:10.48550/arxiv.1503.09034

arxiv created 2015/03/31 · arxiv updated 2015/04/01

Abstract

We prove that the Besicovitch Covering Property (BCP) does not hold for some classes of homogeneous quasi-distances on Carnot groups of step 3 and higher. As a special case we get that, in Carnot groups of step 3 and higher, BCP is not satisfied for those homogeneous distances whose unit ball centered at the origin coincides with a Euclidean ball centered at the origin. This result comes in constrast with the case of the Heisenberg groups where such distances satisfy BCP.

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