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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
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.

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Additional Information

Dmitri Burago
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Sergei Ivanov
Affiliation: St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

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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