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Consider a collection $C_1,C_2, \dots, C_n$ of circles in the plane and suppose that the center of $C_i$ is $o_i$ and the radius of $C_i$ is $r_i$. We will define the relative distance between the circles $C_i$ and $C_j$ as $$\frac {d(o_1,o_2)}{r_1+r_2},$$ where $d(x,y)$ is the Euclidean distance. So the relative distance is $\le 1$ if the two circles are intersecting, and $\ge 1$ if they are non overlapping.

What is the smallest possible number of distinct relative distances?

For unit circles this is Erdos' distinct distances problem that was famously settled by Guth and Katz.

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  • $\begingroup$ Are you ranging over choices of $r_i$ or are the $r_i$ part of the input to the question? $\endgroup$ – Sam Hopkins Feb 6 at 23:20
  • $\begingroup$ Dear @SamHopkins, The "smallest possible number" is a function of $n$ and the input is an arbitrary collections of $n$ circuits. (Each circuit is determined by its center and radius.) $\endgroup$ – Gil Kalai Feb 7 at 8:14
  • $\begingroup$ I am not sure that the relative distance is the most appropriate measure to take. Another option would be to take the cross ratio between the four points $o_i, x_{ij}, y_{ij},o_j$ where $x_{ij}$ and $y_{ij}$ are the intersections of the line segment $[o_i,o_j]$ with $C_i$ and $C_j$ respectively. $\endgroup$ – Gil Kalai Feb 9 at 6:33
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    $\begingroup$ Do you have an example when it can be less than for points? oeis.org/A186704 $\endgroup$ – domotorp Feb 19 at 21:55
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    $\begingroup$ @domotorp, Oops sorry we do have 4 (pairwise tangent) circles in the plane of relative distance 1. I dont know if other relative distances allow more. So it is less than for points. $\endgroup$ – Gil Kalai yesterday

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