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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions
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by Ramón J. Aliaga, Chris Gartland, Colin Petitjean and Antonín Procházka PDF
Trans. Amer. Math. Soc. 375 (2022), 3529-3567 Request permission


We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. We subsequently use this geometric characterization to answer several questions in Lipschitz analysis. Notably, it follows that the Lipschitz-free space $\mathcal {F}(M)$ over a compact metric space $M$ is a dual space if and only if $M$ is purely 1-unrectifiable. Furthermore, we establish a compact determinacy principle for the Radon-Nikodým property (RNP) and deduce that, for any complete metric space $M$, pure 1-unrectifiability is actually equivalent to some well-known Banach space properties of $\mathcal {F}(M)$ such as the RNP and the Schur property. A direct consequence is that any complete, purely 1-unrectifiable metric space isometrically embeds into a Banach space with the RNP. Finally, we provide a possible solution to a problem of Whitney by finding a rectifiability-based description of 1-critical compact metric spaces, and we use this description to prove the following: a bounded turning tree fails to be 1-critical if and only if each of its subarcs has $\sigma$-finite Hausdorff 1-measure.
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Additional Information
  • Ramón J. Aliaga
  • Affiliation: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera S/N, 46022 Valencia, Spain
  • ORCID: 0000-0002-2513-7711
  • Email:
  • Chris Gartland
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
  • MR Author ID: 1238876
  • Email:
  • Colin Petitjean
  • Affiliation: LAMA, Univ. Gustave Eiffel, UPEM, Univ. Paris Est Creteil, CNRS, F-77447 Marne-la-Vallée, France
  • MR Author ID: 1212067
  • ORCID: 0000-0002-6584-5405
  • Email:
  • Antonín Procházka
  • Affiliation: Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, CNRS UMR-6623, 16, route de Gray, 25030 Besançon Cedex, France
  • ORCID: 0000-0001-7762-9147
  • Email:
  • Received by editor(s): August 4, 2021
  • Received by editor(s) in revised form: October 29, 2021
  • Published electronically: February 17, 2022
  • Additional Notes: The first author was partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under Grant MTM2017-83262-C2-2-P. The third and fourth authors were partially supported by the French ANR project No. ANR-20-CE40-0006
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 3529-3567
  • MSC (2020): Primary 51F30; Secondary 28A78, 30L05, 46B20, 46B22, 54E45
  • DOI:
  • MathSciNet review: 4402669