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Sufficient Condition for Rectifiability Involving Wasserstein Distance \(W_2\)

Abstract

A Radon measure \(\mu \) is n-rectifiable if it is absolutely continuous with respect to \({\mathcal {H}}^n\) and \(\mu \)-almost all of \({{\,\mathrm{supp}\,}}\mu \) can be covered by Lipschitz images of \({\mathbb {R}}^n\). In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called \(\alpha \) and \(\beta _2\) numbers. The second one involves \(\alpha _2\) numbers—coefficients quantifying flatness via Wasserstein distance \(W_2\). Both conditions are necessary for rectifiability, too—the first one was shown to be necessary by Tolsa, while the necessity of the \(\alpha _2\) condition is established in our recent paper. Thus, we get two new characterizations of rectifiability.

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Acknowledgements

The author would like to express his deep gratitude to Xavier Tolsa for all his help and guidance. He acknowledges the support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445), and also partial support from the Catalan Agency for Management of University and Research Grants (2017-SGR-0395), and from the Spanish Ministry of Science, Innovation and Universities (MTM-2016-77635-P).

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Correspondence to Damian Dąbrowski.

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Dąbrowski, D. Sufficient Condition for Rectifiability Involving Wasserstein Distance \(W_2\). J Geom Anal 31, 8539–8606 (2021). https://doi.org/10.1007/s12220-020-00603-y

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Keywords

  • Rectifiability
  • Rectifiable measures
  • \(\alpha \) numbers
  • \(\beta \) numbers

Mathematics Subject Classification

  • 28A75
  • 28A78