Packing and covering with balls on Busemann surfaces

TL;DR

This paper establishes a bound relating the minimum number of balls needed to cover a compact set in Busemann surfaces to the maximum number of disjoint balls that can be packed within it, with a universal constant factor.

Contribution

It proves a universal constant factor bound between covering and packing numbers for sets in Busemann surfaces, including simple polygons with geodesic metrics.

Findings

Covering number is at most 19 times the packing number.

The bound applies to any compact subset of Busemann surfaces.

Includes simple polygons with geodesic metrics.

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.