Revista Matemática Iberoamericana

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Volume 35, Issue 5, 2019, pp. 1485–1500
DOI: 10.4171/rmi/1089

Published online: 2019-06-04

A comparison of Euclidean and Heisenberg Hausdorff measures

Pertti Mattila[1] and Laura Venieri[2]

(1) University of Helsinki, Finland
(2) University of Helsinki, Finland

We prove some geometric properties of sets in the first Heisenberg group whose Heisenberg Hausdorff dimension is the minimal or maximal possible in relation to their Euclidean one and the corresponding Hausdorff measures are positive and finite. In the first case we show that these sets must be in a sense horizontal and in the second case vertical. We show the sharpness of our results with some examples.

Keywords: Hausdorff measure, Heisenberg group, Hausdorff dimension

Mattila Pertti, Venieri Laura: A comparison of Euclidean and Heisenberg Hausdorff measures. Rev. Mat. Iberoam. 35 (2019), 1485-1500. doi: 10.4171/rmi/1089