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Discrete & Computational Geometry

, Volume 53, Issue 1, pp 16–37 | Cite as

Isometric Embedding of Busemann Surfaces into \(L_1\)

  • Jérémie Chalopin
  • Victor Chepoi
  • Guyslain NavesEmail author
Article

Abstract

In this paper, we prove that any non-positively curved 2-dimensional surface (alias, Busemann surface) is isometrically embeddable into \(L_1\). As a corollary, we obtain that all planar graphs which are 1-skeletons of planar non-positively curved complexes with regular Euclidean polygons as cells are \(L_1\)-embeddable with distortion at most \(2\). Our results significantly improve and simplify the results of the recent paper by A. Sidiropoulos (Non-positive curvature and the planar embedding conjecture, FOCS (2013)).

Keywords

Isometric embedding Planar embedding CAT(0) surfaces  Non-positive curvature Distortion 

Notes

Acknowledgments

The authors would like to thank James Lee for pointing out an error in the formulation of Theorem 2 in the first version of the article. They also thank the referee for careful reading and useful remarks.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Jérémie Chalopin
    • 1
  • Victor Chepoi
    • 1
  • Guyslain Naves
    • 1
    Email author
  1. 1.Laboratoire d’Informatique Fondamentale, Aix-Marseille Université and CNRS, Faculté des Sciences de LuminyMarseille Cedex 9France

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