In Why is a circle in a plane surrounded by 6 other circles?, the implicit assupmtion is the distance is Euclidean, my question is: Are there any relation between the distance function being used and the number of circles required to surrond a circle? for example
1.Given the number of circles required to surrond a circle what can be deduced about the distance function?
2.Given a distance function, is there any way to calculate the number of circles required to surround a circle?
Addendum : for example consider the taxi cab metric, how many circle does it take to suround a circle in that metric? conversly is there a metric that would cause that n circles be required to suround the circle, given n find the metric.
Note : We are not dealing with (non-)Euclidean geometries, but with different metrics induced on plane ( although some metrics could be isometric to being on Sphere etc. ) It is a question with planer geometry using different distance functions to induce a metric space.
