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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.

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