Packing and covering with balls on Busemann surfaces

Abstract

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.

27 pages

References in corpus (2)

• Packing and Covering a Polygon with Geodesic Disks
• Isometric embedding of Busemann surfaces into $L_1$

Cited by in corpus (1)

• Packing and covering balls in graphs excluding a minor