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Isometric approximation

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Abstract

Suppose thatA ⊂R n is a bounded set of diameter 1 and that:f:A →l 2 is a map satisfying the nearisometry condition |x−y|−ɛ≤|fx−fy|≤|x−y|+ɛ withɛ≤1. Then there is an isometryS:A →l 2 such that |Sx−fx|≤c n √ɛ for allx inA. IfA satisfies a thickness condition and iff:A →R n, then there is an isometryS:R n →R n with |Sx−fx|≤c nɛ/q, whereq is a thickness parameter.

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References

  1. Y. Benyamini and J. Lindenstrauss,Geometric nonlinear functional analysis I, American Mathematical Society Colloquium Publications48 (2000).

  2. D. H. Hyers and S. M. Ulam,On approximate isometries, Bulletin of the American Mathematical Society51 (1945), 288–292.

    MATH  MathSciNet  Google Scholar 

  3. J. Fickett,Approximate isometries on bounded sets with an application to measure theory, Studia Mathematica72 (1982), 37–46.

    MATH  MathSciNet  Google Scholar 

  4. J. Gevirtz,Stability of isometries on Banach spaces, Proceedings of the American Mathematical Society89 (1983), 633–636.

    Article  MATH  MathSciNet  Google Scholar 

  5. F. John,Rotation and strain, Communications on Pure and Applied Mathematics14 (1961), 391–413.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Omladič and P. Šemrl,On nonlinear perturbations of isometries, Mathematische Annalen303 (1995), 617–628.

    Article  Google Scholar 

  7. P. Šemrl,Hyers-Ulam stability of isometries, Houston Journal of Mathematics24 (1998), 699–706.

    MathSciNet  Google Scholar 

  8. D. A. Trotsenko,Approximation of maps of bounded distortion by similarities, Sibirskii Matematicheskii Zhurnal27 (1986), 196–205 (Russian).

    MathSciNet  Google Scholar 

  9. P. Tukia and J. Väisälä,Quasisymmetric embeddings of metric spaces, Annales Academiae Scientiarum Fennicae. Mathematica5 (1980), 97–114.

    Google Scholar 

  10. J. Väisälä,Bilipschitz and quasisymmetric extension properties, Annales Academiae Scientiarum Fennicae. Mathematica11 (1986), 239–274.

    MATH  Google Scholar 

  11. J. Väisälä, M. Vuorinen and H. Wallin,Thick sets and quasisymmetric maps, Nagoya Mathematical Journal135 (1994), 121–148.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Matematiikan laitos, Teknillinen korkeakoulu, PL 1100, 02015, TKK, Finland

    P. Alestalo

  2. Institut Matematiki SO RAN, Koptjuga prospekt 4, 630090, Novosibirsk, Russia

    D. A. Trotsenko

  3. Matematiikan laitos, Helsingin yliopisto, PL 4, Yliopistonkatu 5, 00014, Helsinki, Finland

    J. Väisälä

Authors
  1. P. Alestalo
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  2. D. A. Trotsenko
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  3. J. Väisälä
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Correspondence to P. Alestalo.

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Alestalo, P., Trotsenko, D.A. & Väisälä, J. Isometric approximation. Isr. J. Math. 125, 61–82 (2001). https://doi.org/10.1007/BF02773375

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  • DOI: https://doi.org/10.1007/BF02773375

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