In this note we prove that for any compact subset S of a Busemann surface @math (S,d) (in particular, for any simple polygon with geodesic metric) and any positive number @math ź, the minimum number of closed balls of radius @math ź with centers at @math S and covering the set S is at most 19 times the maximum number of disjoint closed balls of radius @math ź centered at points of S: @math ź(S)≤ź(S)≤19ź(S), where @math ź(S) and @math ź(S) are the covering and the packing numbers of S by @math ź-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 @math R2 in which the distance function is convex.
Given a polygon @math , for two points @math and @math contained in the polygon, their is the length of the shortest @math -path within @math . A of radius @math centered at a point @math is the set of points in @math whose geodesic distance to @math is at most @math . We present a polynomial time @math -approximation algorithm for finding a densest geodesic unit disk packing in @math . Allowing arbitrary radii but constraining the number of disks to be @math , we present a @math -approximation algorithm for finding a packing in @math with @math geodesic disks whose minimum radius is maximized. We then turn our focus on of @math and present a @math -approximation algorithm for covering @math with @math geodesic disks whose maximal radius is minimized. Furthermore, we show that all these problems are @math -hard in polygons with holes. Lastly, we present a polynomial time exact algorithm which covers a polygon with two geodesic disks of minimum maximal radius. The geodesic center of a simple polygon is a point inside the polygon which minimizes the maximum internal distance to any point in the polygon. We present an algorithm which calculates the geodesic center of a simple polygon withn vertices in timeO(n logn).