Definitions: Let $D_k$ be the disk with constant curvature $k$ and with volume $1$. We call a finite subset $X\subset D_k$ a $\delta$-covering of $D_k$ if $D_k=\bigcup_{x\in X}B_\delta(x)$, where $B_\delta(x)=(p\in D_k \vert\,d(p,x) \leq \delta)$ is the closed $\delta$-ball in $D_k$ with respect to the geodesic distance.
Question: What is the minimum number of $\delta$-balls required to cover $D_k$, that is $$ N_{\delta,k} = \text{min}_{X \text{ is }\delta\text{-covering of }D_k} \vert X\vert. $$ In the euclidean case $(k=0)$ this seems to be an old and answered question, but what about the hyperbolic and spherical case? For the spherical case I have found some work concerned with minimizing covering density, which is the average number of balls containing a point.
Upper bounds can be found by constructing $\delta$-coverings. In the euclidean case, the best possible configuration is to put the centers of the balls on a hexagonal grid. This solution scales nicely in $\delta$. However, in non-euclidean space I do not know of a tiling which is scalable like the hexagonal grid in the euclidean plane. For instance the Poincare disk $D_{-1}$ can be tiled by equilateral triangles, resulting in a grid of order 7. However this only works for triangles with a specific side-length.