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Article Dans Une Revue Discrete and Computational Geometry Année : 2017

Packing and Covering with Balls on Busemann Surfaces

Résumé

In this note we prove that for any compact subset S of a Busemann surface (S, d) (in particular, for any simple polygon with geodesic metric) and any positive number δ, the minimum number of closed balls of radius δ with centers at S and covering the set S is at most 19 times the maximum number of disjoint closed balls of radius δ centered at points of S: ν(S) ≤ ρ(S) ≤ 19ν(S), where ρ(S) and ν(S) are the covering and the packing numbers of S by δ-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 R 2 in which the distance function is convex.
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Dates et versions

hal-03581534 , version 1 (06-10-2023)

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Victor Chepoi, Bertrand Estellon, Guyslain Naves. Packing and Covering with Balls on Busemann Surfaces. Discrete and Computational Geometry, 2017, 57, pp.985 - 1011. ⟨10.1007/s00454-017-9872-0⟩. ⟨hal-03581534⟩
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