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


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)).


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



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.


  1. 1.
    Alexander, R.: Planes for which the lines are the shortest paths between points. Ill. J. Math. 22, 177–190 (1978)zbMATHGoogle Scholar
  2. 2.
    Bandelt, H.-J., Chepoi, V.: Metric graph theory and geometry: a survey. In: Goodman, J.E., Pach, J., Pollack, R. (eds.) Surveys on Discrete and Computational Geometry. Twenty Years Later. Contemporary Mathematics, vol. 453, pp. 49–86. AMS, Providence, RI (2008)CrossRefGoogle Scholar
  3. 3.
    Baues, O., Peyerimhoff, N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25, 141–159 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bourgain, J.: On Lipschitz embedding of finite metric spaces in Hilbert space. Isr. J. Math. 52, 46–52 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bretagnolle, J., Dacunha Castelle, D., Krivine, J.-L.: Lois stables et espaces \(L_p\). Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 2, 231–259 (1966)Google Scholar
  6. 6.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlehren Math. Wiss., vol. 319. Springer, Berlin (1999)CrossRefGoogle Scholar
  7. 7.
    Chepoi, V., Dragan, F., Vaxès, Y.: Distance and routing problems in plane graphs of non-positive curvature. J. Algorithms 61, 1–30 (2006)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Chepoi, V., Fichet, B.: A note on circular decomposable metrics. Geom. Dedicata 69, 237–240 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Algorithms Comb., vol. 15. Springer, Berlin (1997)Google Scholar
  10. 10.
    Dhandapani, R., Goodman, J.E., Holmsen, A., Pollack, R., Smorodinsky, S.: Convexity in topological affine planes. Discrete Comput. Geom. 38, 243–257 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Foertsch, T., Lytchak, A., Schroeder, V. : Non-positive curvature and the Ptolemy inequality. Int. Math. Res. Not. (2007). doi: 10.1093/imrn/rnm100
  12. 12.
    Gromov, M.: Hyperbolic goups. In: Gersten, S.M. (ed.) Essays in Group Theory. MSRI Series, vol. 8. Springer, Berlin (1987)Google Scholar
  13. 13.
    Gupta, A., Newman, I., Rabinovich, Y., Sinclair, A.: Cuts, trees and \(l_1\)-embeddings of graphs. Combinatorica 24, 233–269 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Indyk, P., Matoušek, J.: Low-distortion embeddings of finite metric spaces. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn, pp. 177–196. CRC Press LLC, Boca Raton (2004)Google Scholar
  15. 15.
    Maftuleac, D.: Algorithmique des complexes CAT(0) planaires et rectangulaires. Ph.D. thesis, Aix-Marseille Université. (2012)
  16. 16.
    Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Okamura, H., Seymour, P.D.: Multicommodity flows in planar graphs. J. Comb. Theory, Ser. B 31, 75–81 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Papadopulos, A.: Metric Spaces, Convexity and Nonpositive Curvature, IRMA Lectures in Mathematics and Theoretical Physics, vol. 6. European Mathematical Society, Zurich (2005)Google Scholar
  19. 19.
    Sidiropoulos, A.: Non-positive curvature, and the planar embedding conjecture. FOCS 2013, pp. 177–186. (2013)
  20. 20.
    van de Vel, M.: Theory of Convex Structures. Elsevier, Amsterdam (1993)zbMATHGoogle Scholar

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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