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Uniform approximation of metrics by graphs


Authors: Dmitri Burago and Sergei Ivanov
Journal: Proc. Amer. Math. Soc. 143 (2015), 1241-1256
MSC (2010): Primary 51K05, 05C12
DOI: https://doi.org/10.1090/S0002-9939-2014-12299-4
Published electronically: October 16, 2014
MathSciNet review: 3293739
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Abstract: We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one within a finite Gromov-Hausdorff distance. We show that the Euclidean plane and Gromov hyperbolic geodesic spaces with bounded geometry are approximable by uniform graphs, and pose a number of open problems.


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

  • [1] József Beck and William W. L. Chen, Irregularities of distribution, Cambridge Tracts in Mathematics, vol. 89, Cambridge University Press, Cambridge, 1987. MR 903025 (88m:11061)
  • [2] Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418 (2002e:53053)
  • [3] J. Pach, R. Pollack, and J. Spencer, Graph distance and Euclidean distance on the grid, Topics in combinatorics and graph theory (Oberwolfach, 1990) Physica, Heidelberg, 1990, pp. 555-559. MR 1100078 (91m:05077)
  • [4] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776 (21 #6498)

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Additional Information

Dmitri Burago
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: burago@math.psu.edu

Sergei Ivanov
Affiliation: St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: svivanov@pdmi.ras.ru

DOI: https://doi.org/10.1090/S0002-9939-2014-12299-4
Keywords: Metric graph, Gromov-Hausdorff distance
Received by editor(s): October 12, 2012
Received by editor(s) in revised form: June 25, 2013
Published electronically: October 16, 2014
Additional Notes: The first author was partially supported by NSF grant DMS-1205597.
The second author was partially supported by RFBR grant 11-01-00302-a.
Communicated by: Kevin Whyte
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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