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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel


IMPACT FACTOR 2018: 0.536

CiteScore 2018: 0.83

SCImago Journal Rank (SJR) 2018: 1.041
Source Normalized Impact per Paper (SNIP) 2018: 0.801

Mathematical Citation Quotient (MCQ) 2018: 0.83

Open Access
Online
ISSN
2299-3274
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Products of Snowflaked Euclidean Lines Are Not Minimal for Looking Down

Matthieu Joseph / Tapio Rajala
  • Corresponding author
  • University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyvaskyla, Jyvaskyla, Finland
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Published Online: 2017-11-16 | DOI: https://doi.org/10.1515/agms-2017-0005

Abstract

We show that products of snowflaked Euclidean lines are not minimal for looking down. This question was raised in Fractured fractals and broken dreams, Problem 11.17, by David and Semmes. The proof uses arguments developed by Le Donne, Li and Rajala to prove that the Heisenberg group is not minimal for looking down. By a method of shortcuts, we define a new distance d such that the product of snowflaked Euclidean lines looks down on (RN , d), but not vice versa.

Keywords: Ahlfors-regularity; biLipschitz pieces; BPI-spaces

MSC 2010: Primary 26B05; Secondary 28A80

References

  • [1] Luigi Ambrosio and Paolo Tilli, Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, 2004.Google Scholar

  • [2] Andrew M. Bruckner, Judith B. Bruckner, and Brian S. Thomson, Real analysis, Prentice-Hall, 1997.Google Scholar

  • [3] Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, American Mathematical Society, 2001.Google Scholar

  • [4] Guy David and Stephen Semmes, Fractured fractals and broken dreams: self-similar geometry through metric and measure, Oxford Lecture Series in Mathematics and Its Applications 7, Clarendon Press, 1997.Google Scholar

  • [5] Bernd Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), 113-123.Google Scholar

  • [6] Tomi J. Laakso, Look-down equivalence without BPI equivalence, (2002), Preprint.Google Scholar

  • [7] Enrico Le Donne, Sean Li, and Tapio Rajala, Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces, Proc. Lond. Math. Soc. (3) 115 (2017), no. 2, 348-380. MR 3684108Google Scholar

About the article

Received: 2017-08-16

Accepted: 2017-10-24

Published Online: 2017-11-16


Citation Information: Analysis and Geometry in Metric Spaces, Volume 5, Issue 1, Pages 78–97, ISSN (Online) 2299-3274, DOI: https://doi.org/10.1515/agms-2017-0005.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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