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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0213-0,
author = {Stephen Buckley and Safia Hamza},
title = {Best constants for metric space inversion inequalities},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {865-875},
zbl = {1288.30065},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0213-0}
}
Stephen Buckley; Safia Hamza. Best constants for metric space inversion inequalities. Open Mathematics, Tome 11 (2013) pp. 865-875. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0213-0/
[1] Bonk M., Heinonen J., Koskela P., Uniformizing Gromov Hyperbolic Spaces, Astérisque, 207, Société Mathématique de France, Paris, 2001
[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 | Zbl 1171.53024
[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 | Zbl 1160.30006
[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 | Zbl 1203.53030
[5] Foertsch T., Lytchak A., Schroeder V., Nonpositive curvature and the Ptolemy inequality, Int. Math. Res. Not. IMRN, 2007, 22, #rnm100 | Zbl 1135.53055
[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 | Zbl 1219.53042
[7] Herron D., Shanmugalingam N., Xie X., Uniformity from Gromov hyperbolicity, Illinois J. Math., 2008, 52(4), 1065–1109 | Zbl 1189.30055