Skip to main content
SpringerLink
Log in

Boundedness of precompact sets of metric measure spaces

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We give a detailed proof to Gromov’s statement that precompact sets of metric measure spaces are bounded with respect to the box distance and the Lipschitz order.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Not applicable.

References

  1. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  2. Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152. Birkhäuser Boston Inc., Boston (1999)

    Google Scholar 

  3. Kazukawa, D.: Convergence of metric transformed spaces. arXiv:2008.04038

  4. Nakajima, H.: Isoperimetric inequality on a metric measure space and Lipschitz order with an additive error. arXiv:1902.07424

  5. Ozawa, R., Yokota, T.: Stability of RCD condition under concentration topology. Calc. Var. Partial Differ. Equ. 58(4), 151 (2019)

    Article  MathSciNet  Google Scholar 

  6. Shioya, T.: Metric measure geometry. Gromov’s theory of convergence and concentration of metrics and measures. In: IRMA Lectures in Mathematics and Theoretical Physics, vol. 25. EMS Publishing House, Zürich (2016)

  7. Tuzhilin, A.A.: Lectures on Hausdorff and Gromov–Hausdorff Distance Geometry. arXiv:2012.00756

Download references

Acknowledgements

The authors thank Professor Takashi Shioya and the referee for their comments. The first author was partly supported by JSPS KAKENHI (No. 20J00147) and the second author was partly supported by JSPS KAKENHI (No. 18K03298).

Author information

Authors and Affiliations

  1. Department of Mathematics, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Daisuke Kazukawa

  2. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Takumi Yokota

Authors
  1. Daisuke Kazukawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takumi Yokota
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daisuke Kazukawa.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kazukawa, D., Yokota, T. Boundedness of precompact sets of metric measure spaces. Geom Dedicata 215, 229–242 (2021). https://doi.org/10.1007/s10711-021-00646-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-021-00646-7

Keywords

2020 Mathematics Subject Classification

Search

Navigation