Bi-Lipschitz embedding of projective metrics

Kovalev, Leonid

doi:10.1090/S1088-4173-2014-00266-5

Bi-Lipschitz embedding of projective metrics

Author: Leonid V. Kovalev
Journal: Conform. Geom. Dyn. 18 (2014), 110-118
MSC (2010): Primary 30L05; Secondary 30C65, 51M10
DOI: https://doi.org/10.1090/S1088-4173-2014-00266-5
Published electronically: June 6, 2014
MathSciNet review: 3215428
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a sufficient condition for a projective metric on a subset of a Euclidean space to admit a bi-Lipschitz embedding into Euclidean space of the same dimension.

[1] Ralph Alexander, Planes for which the lines are the shortest paths between points, Illinois J. Math. 22 (1978), no. 2, 177–190. MR 490820
[2] Juan C. Álvarez Paiva, Hilbert’s fourth problem in two dimensions, MASS selecta, Amer. Math. Soc., Providence, RI, 2003, pp. 165–183. MR 2027175
[3] R. V. Ambartzumian, Combinatorial solution of the Buffon-Sylvester problem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 29 (1974), 25–31. MR 372939, https://doi.org/10.1007/BF00533183
[4] Yves Benoist, Convexes hyperboliques et fonctions quasisymétriques, Publ. Math. Inst. Hautes Études Sci. 97 (2003), 181–237 (French, with English summary). MR 2010741, https://doi.org/10.1007/s10240-003-0012-4
[5] A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142. MR 86869, https://doi.org/10.1007/BF02392360
[6] Herbert Busemann, Geometries in which the planes minimize area, Ann. Mat. Pura Appl. (4) 55 (1961), 171–189. MR 143155, https://doi.org/10.1007/BF02412083
[7] Herbert Busemann, Problem IV: Desarguesian spaces, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 131–141. Proc. Sympos. Pure Math., Vol. XXVIII. MR 0430935
[8] Herbert Busemann and Paul J. Kelly, Projective geometry and projective metrics, Academic Press Inc., New York, N. Y., 1953. MR 0054980
[9] Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917
[10] Tomi J. Laakso, Plane with 𝐴_{∞}-weighted metric not bi-Lipschitz embeddable to ℝ^{ℕ}, Bull. London Math. Soc. 34 (2002), no. 6, 667–676. MR 1924353, https://doi.org/10.1112/S0024609302001200
[11] Anders Karlsson and Guennadi A. Noskov, The Hilbert metric and Gromov hyperbolicity, Enseign. Math. (2) 48 (2002), no. 1-2, 73–89. MR 1923418
[12] Leonid V. Kovalev, Quasiconformal geometry of monotone mappings, J. Lond. Math. Soc. (2) 75 (2007), no. 2, 391–408. MR 2340234, https://doi.org/10.1112/jlms/jdm008
[13] Leonid Kovalev, Diego Maldonado, and Jang-Mei Wu, Doubling measures, monotonicity, and quasiconformality, Math. Z. 257 (2007), no. 3, 525–545. MR 2328811, https://doi.org/10.1007/s00209-007-0132-5
[14] Athanase Papadopoulos, On Hilbert's fourth problem, available at arXiv:1312.3172.
[15] Aleksei Vasil′evich Pogorelov, Hilbert’s fourth problem, V. H. Winston & Sons, Washington, D.C.; A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont.-London, 1979. Translated by Richard A. Silverman; Scripta Series in Mathematics. MR 550440
[16] R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original; Princeton Paperbacks. MR 1451876
[17] Rolf Schneider and Wolfgang Weil, Stochastic and integral geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. MR 2455326
[18] Stephen Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about 𝐴_{∞}-weights, Rev. Mat. Iberoamericana 12 (1996), no. 2, 337–410. MR 1402671, https://doi.org/10.4171/RMI/201
[19] Z. I. Szabó, Hilbert’s fourth problem. I, Adv. in Math. 59 (1986), no. 3, 185–301. MR 835025, https://doi.org/10.1016/0001-8708(86)90056-3
[20] Jussi Väisälä, Lectures on 𝑛-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009