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A remark on quasi-isometries


Author: N. J. Kalton
Journal: Proc. Amer. Math. Soc. 131 (2003), 1225-1231
MSC (2000): Primary 46C05, 47H99
DOI: https://doi.org/10.1090/S0002-9939-02-06663-7
Published electronically: July 26, 2002
MathSciNet review: 1948114
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Abstract: We show that if $f:B_n\to\mathbb R^n$ is an $\epsilon-$quasi-isometry, with $\epsilon<1$, defined on the unit ball $B_n$ of $\mathbb R^n$, then there is an affine isometry $h:B_n\to\mathbb R^n$ with $\Vert f(x)-h(x)\Vert\le C\epsilon (1+\log n)$ where $C$ is a universal constant. This result is sharp.


References [Enhancements On Off] (What's this?)

  • 1. P. Alestalo, D.A. Trotsenko and J. Väisaäla, Isometric approximation, Israel J. Math. 125 (2001) 61-82.
  • 2. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Volume 1 Colloquium Publications No. 48, Amer. Math. Soc. 2000. MR 2001b:46001
  • 3. Y. Brudnyi and N.J. Kalton, Polynomial approximation on convex subsets of $\bf R^n,$ Constructive Approximation 16 (2000) 161-200. MR 2000k:41036
  • 4. P. Enflo, J. Lindenstrauss and G. Pisier, On the three space problem, Math. Scand. 36 (1975) 199-210. MR 52:3928
  • 5. T. Figiel, J. Lindenstrauss and V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta. Math. 139 (1977) 53-94. MR 56:3618
  • 6. F. John, Rotation and strain, Comm. Pure. Appl. Math. 14 (1961) 391-413. MR 25:1672
  • 7. N.J. Kalton and N.T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979) 1-30. MR 82g:46021
  • 8. E. Matouskova, Almost isometries of balls, J. Functional Analysis 190 (2002) 507-525.
  • 9. B. Maurey, Une théorème de prolongement, C.R. Acad. Sci Paris 279 (1974) 329-332. MR 50:8013
  • 10. V.I. Semenov, Estimates of stability for spatial quasiconfromal mappings of a starlike region, Siberian Math. J. 28 (1987) 946-955.
  • 11. I.A. Vestfrid, Affine properties and injectivity of quasi-isometries, to appear.
  • 12. H. Whitney, On functions with bounded $n$th differences, J. Math. Pures Appl. 36 (1957) 67-95. MR 18:889f

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Additional Information

N. J. Kalton
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
Email: nigel@math.missouri.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06663-7
Keywords: Quasi-isometries in Euclidean spaces
Received by editor(s): June 10, 2001
Received by editor(s) in revised form: November 27, 2001
Published electronically: July 26, 2002
Additional Notes: The author was supported by NSF grant DMS-9870027
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2002 American Mathematical Society

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