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Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones

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Abstract

We prove that if there exists a bi-Lipschitz homeomorphism (not necessarily subanalytic) between two subanalytic sets, then their tangent cones are bi-Lipschitz homeomorphic. As a consequence of this result, we show that any Lipschitz regular complex analytic set, i.e., any complex analytic set which is locally bi-Lipschitz homeomorphic to an Euclidean ball must be smooth. Finally, we give an alternative proof of S. Koike and L. Paunescu’s result about the bi-Lipschitz invariance of directional dimensions of subanalytic sets.

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Acknowledgments

I wish to thank my thesis advisor Alexandre Fernandes and Lev Birbrair by incentive in this research and also for their support and help. I also wish to thank Vincent Grandjean and anonymous referee for corrections and suggestions in writing this article. The result of the paper is part of my PhD thesis at Universidade Federal do Ceará and the author was partially supported by FUNCAP.

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  1. Departamento de Matemática, Universidade Federal do Ceará, Av. Humberto Monte, s/n Campus do Pici - Bloco 914, Fortaleza, CE, 60455-760, Brazil

    J. Edson Sampaio

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  1. J. Edson Sampaio
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Correspondence to J. Edson Sampaio.

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Sampaio, J.E. Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones. Sel. Math. New Ser. 22, 553–559 (2016). https://doi.org/10.1007/s00029-015-0195-9

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