Best constants for metric space inversion inequalities

Stephen Buckley; Safia Hamza

Open Mathematics (2013)

  • Volume: 11, Issue: 5, page 865-875
  • ISSN: 2391-5455

Abstract

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For every metric space (X, d) and origin o ∈ X, we show the inequality I o(x, y) ≤ 2d o(x, y), where I o(x, y) = d(x, y)/d(x, o)d(y, o) is the metric space inversion semimetric, d o is a metric subordinate to I o, and x, y ∈ X o The constant 2 is best possible.

How to cite

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Stephen Buckley, and Safia Hamza. "Best constants for metric space inversion inequalities." Open Mathematics 11.5 (2013): 865-875. <http://eudml.org/doc/269794>.

@article{StephenBuckley2013,
abstract = {For every metric space (X, d) and origin o ∈ X, we show the inequality I o(x, y) ≤ 2d o(x, y), where I o(x, y) = d(x, y)/d(x, o)d(y, o) is the metric space inversion semimetric, d o is a metric subordinate to I o, and x, y ∈ X o The constant 2 is best possible.},
author = {Stephen Buckley, Safia Hamza},
journal = {Open Mathematics},
keywords = {Metric space inversion; Best constant; metric space; inversion; isometry},
language = {eng},
number = {5},
pages = {865-875},
title = {Best constants for metric space inversion inequalities},
url = {http://eudml.org/doc/269794},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Stephen Buckley
AU - Safia Hamza
TI - Best constants for metric space inversion inequalities
JO - Open Mathematics
PY - 2013
VL - 11
IS - 5
SP - 865
EP - 875
AB - For every metric space (X, d) and origin o ∈ X, we show the inequality I o(x, y) ≤ 2d o(x, y), where I o(x, y) = d(x, y)/d(x, o)d(y, o) is the metric space inversion semimetric, d o is a metric subordinate to I o, and x, y ∈ X o The constant 2 is best possible.
LA - eng
KW - Metric space inversion; Best constant; metric space; inversion; isometry
UR - http://eudml.org/doc/269794
ER -

References

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  1. [1] Bonk M., Heinonen J., Koskela P., Uniformizing Gromov Hyperbolic Spaces, Astérisque, 207, Société Mathématique de France, Paris, 2001 
  2. [2] Buckley S.M., Falk K., Wraith D.J., Ptolemaic spaces and CAT(0), Glasg. Math. J., 2009, 51(2), 301–314 http://dx.doi.org/10.1017/S0017089509004984 Zbl1171.53024
  3. [3] Buckley S.M., Herron D.A., Xie X., Metric space inversions, quasihyperbolic distance, and uniform spaces, Indiana Univ. Math. J., 2008, 57(2), 837–890 Zbl1160.30006
  4. [4] Buckley S.M., Wraith D.J., McDougall J., On Ptolemaic metric simplicial complexes, Math. Proc. Cambridge Philos. Soc., 2010, 149(1), 93–104 http://dx.doi.org/10.1017/S0305004110000125 Zbl1203.53030
  5. [5] Foertsch T., Lytchak A., Schroeder V., Nonpositive curvature and the Ptolemy inequality, Int. Math. Res. Not. IMRN, 2007, 22, #rnm100 Zbl1135.53055
  6. [6] Foertsch T., Schroeder V., Hyperbolicity, CAT(−1)-spaces and the Ptolemy inequality, Math. Ann., 2011, 350(2), 339–356 http://dx.doi.org/10.1007/s00208-010-0560-0 Zbl1219.53042
  7. [7] Herron D., Shanmugalingam N., Xie X., Uniformity from Gromov hyperbolicity, Illinois J. Math., 2008, 52(4), 1065–1109 Zbl1189.30055

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