Discrete & Computational Geometry

, Volume 57, Issue 4, pp 985–1011 | Cite as

Packing and Covering with Balls on Busemann Surfaces

  • Victor Chepoi
  • Bertrand Estellon
  • Guyslain NavesEmail author


In this note we prove that for any compact subset S of a Busemann surface \(({{\mathcal {S}}},d)\) (in particular, for any simple polygon with geodesic metric) and any positive number \(\delta \), the minimum number of closed balls of radius \(\delta \) with centers at \({\mathcal {S}}\) and covering the set S is at most 19 times the maximum number of disjoint closed balls of radius \(\delta \) centered at points of S: \(\nu (S)\le \rho (S)\le 19{}\nu (S)\), where \(\rho (S)\) and \(\nu (S)\) are the covering and the packing numbers of S by \({\delta }\)-balls. Busemann surfaces represent a far-reaching generalization not only of simple polygons, but also of Euclidean and hyperbolic planes and of all planar polygonal complexes of global non-positive curvature. Roughly speaking, a Busemann surface is a geodesic metric space homeomorphic to \({{\mathbb {R}}}^2\) in which the distance function is convex.


Packing number Covering numbers Balls Busemann surfaces 



The authors would like to thank the referees of this paper for careful reading of the previous versions and many useful remarks.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Victor Chepoi
    • 1
  • Bertrand Estellon
    • 1
  • Guyslain Naves
    • 1
    Email author
  1. 1.Aix Marseille Université, CNRS, LIF UMR 7279MarseilleFrance

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