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Analysis and Geometry in Metric Spaces

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On Conditions for Unrectifiability of a Metric Space

Piotr Hajłasz
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  • Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA
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/ Soheil Malekzadeh
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  • Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA
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Published Online: 2014-12-17 | DOI: https://doi.org/10.1515/agms-2015-0001


We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

Keywords: geometric measure theory; unrectifiability; metric spaces; Sard theorem; Carnot-Carathéodory spaces

MSC: 49Q15; 53C17


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About the article

Received: 2014-03-07

Accepted: 2014-11-15

Published Online: 2014-12-17

Citation Information: Analysis and Geometry in Metric Spaces, Volume 3, Issue 1, ISSN (Online) 2299-3274, DOI: https://doi.org/10.1515/agms-2015-0001.

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© 2015 Piotr Hajłasz, Soheil Malekzadeh. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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Piotr Hajłasz and Soheil Malekzadeh
International Mathematics Research Notices, 2015, Volume 2015, Number 24, Page 13238

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