Results in Mathematics

, Volume 43, Issue 3–4, pp 359–372 | Cite as

Isometric approximation property of unbounded sets

  • Jussi Väisälä


We give a necessary and sufficient quantitative geometric condition for an unbounded set A ⊂ Rn to have the following property with a given c > 0: For every ε ≥ 0 and for every map f: A → Rn such that Open image in new window , there is an isometry T: A → Rn such that ¦Tx−fx¦ ≤ cε for all x ∈ A.

2000 Mathematics Subject Classification

46C05 46B20 30C65 

Key Words

nearisometry Hyers-Ulam isometric approximation 


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  1. [ATV]
    P. Alestalo, D.A. Trotsenko and J. Väisälä, Isometric approximation, Israel J. Math. 125 (2001), 61–82.MathSciNetMATHCrossRefGoogle Scholar
  2. [BL]
    [BL] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis I, AMS Colloquium Publications 48, 2000.Google Scholar
  3. [BŠ]
    R. Bhatia and P. Semrl, Approximate isometries on Euclidean spaces, Amer. Math. Monthly 104 (1997), 497–504.MathSciNetMATHCrossRefGoogle Scholar
  4. [Da]
    J. Danes, On the radius of a set in a Hubert space, Comment. Math. Univ. Carolin. 25 (1984), 355–362.MathSciNetMATHGoogle Scholar
  5. [Di]
    S.J. Dilworth, Approximate isometries in finite-dimensional normed spaces, Bull. London Math. Soc. 31 (1999), 471–476.MathSciNetMATHCrossRefGoogle Scholar
  6. [HV]
    T. Huuskonen and J. Väisälä, Hyers-Ulam constants of Hubert spaces, Studia Math. 153 (2002), 31–40.MathSciNetMATHCrossRefGoogle Scholar
  7. [HU]
    D.H. Hyers and S.M. Ulam, On approximate isometries, Bull. Amer. Math. Soc. 51 (1945), 288–292.MathSciNetMATHCrossRefGoogle Scholar
  8. [Jo]
    F. John, Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391–413.MATHCrossRefGoogle Scholar
  9. [Ju]
    H.W.E. Jung, Über die kleinste Kugel, die eine räumliche Figur einschliesst, J. Reine Angew. Math. 123 (1901), 241–257.MATHGoogle Scholar
  10. [Ma]
    E. Matoušková, Almost isometries of balls, J. Punct. Anal. 190 (2002) 505–525.Google Scholar
  11. [Qi]
    S. Qian, e-isometric embeddings, Proc. Amer. Math. Soc. 123 (1995), 1797–1803.MathSciNetMATHGoogle Scholar
  12. [Re]
    E.G. Rees, Notes on geometry, Springer, 1983.Google Scholar
  13. [Še]
    P. Šemrl, Hyers-Ulam stability of isometries, Houston J. Math. 24 (1998), 699–706.MathSciNetMATHGoogle Scholar
  14. [Vä1]
    J. Väisälä, Isometric approximation property in euclidean spaces, Israel J. Math. 128 (2002), 1–28.MathSciNetMATHCrossRefGoogle Scholar
  15. [Vä2]
    J. Väisälä, A survey of nearisometries, Report. Univ. Jyväskylä 83 (2001), 305–315.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2003

Authors and Affiliations

  1. 1.Matematiikan laitosHelsingin yliopistoHelsinkiFinland

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