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Frontiers of Mathematics in China

, Volume 14, Issue 2, pp 349–380 | Cite as

Lipschitz Gromov-Hausdorff approximations to two-dimensional closed piecewise flat Alexandrov spaces

  • Xueping LiEmail author
Research Article
  • 7 Downloads

Abstract

We show that a closed piecewise flat 2-dimensional Alexandrov space Σ can be bi-Lipschitz embedded into a Euclidean space such that the embedded image of Σ has a tubular neighborhood in a generalized sense. As an application, we show that for any metric space sufficiently close to Σ in the Gromov-Hausdroff topology, there is a Lipschitz Gromov-Hausdorff approximation.

Keywords

Alexandrov space Gromov-Hausdorff approximation tubular neighborhood 

MSC

53C20 53C99 57N65 

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Notes

Acknowledgements

Part of this work was done during the author’s visit to Capital Normal University, thanks a lot for their hospitality. The author would also like to thank the referees for lots of useful comments, especially the one leads to the simplification of the proof of Lemma 1. The author was in great debt to Prof. Xiaochun Rong, without whose help this work can never get the present form. This work was partially supported by the National Natural Science Foundation of China (Grant No. 11501258) and a foundation from Jiangsu Normal University.

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Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsJiangsu Normal UniversityXuzhouChina

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